Algebra Examples

$\begin{array}{c} x & y \\ 2 & 20 \\ 5 & 15 \\ 15 & 2 \end{array}$

Check if the function rule is linear.

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To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Build a set of equations from the table such that $y = a x + b$ .

$20 = a (2) + b 15 = a (5) + b 2 = a (15) + b$

Calculate the values of $a$ and $b$ .

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Solve for $b$ in $20 = a (2) + b$ .

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Rewrite the equation as $a (2) + b = 20$ .

$a (2) + b = 20$

$15 = a (5) + b$

$2 = a (15) + b$

Move $2$ to the left of $a$ .

$2 a + b = 20$

$15 = a (5) + b$

$2 = a (15) + b$

Subtract $2 a$ from both sides of the equation.

$b = 20 - 2 a$

$15 = a (5) + b$

$2 = a (15) + b$

$b = 20 - 2 a$

$15 = a (5) + b$

$2 = a (15) + b$

Replace all occurrences of $b$ with $20 - 2 a$ in each equation.

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Replace all occurrences of $b$ in $15 = a (5) + b$ with $20 - 2 a$ .

$15 = a (5) + 20 - 2 a$

$b = 20 - 2 a$

$2 = a (15) + b$

Simplify $15 = a (5) + 20 - 2 a$ .

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Simplify the left side.

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Remove parentheses.

$15 = a (5) + 20 - 2 a$

$b = 20 - 2 a$

$2 = a (15) + b$

$15 = a (5) + 20 - 2 a$

$b = 20 - 2 a$

$2 = a (15) + b$

Simplify the right side.

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Simplify $a (5) + 20 - 2 a$ .

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Move $5$ to the left of $a$ .

$15 = 5 a + 20 - 2 a$

$b = 20 - 2 a$

$2 = a (15) + b$

Subtract $2 a$ from $5 a$ .

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = a (15) + b$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = a (15) + b$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = a (15) + b$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = a (15) + b$

Replace all occurrences of $b$ in $2 = a (15) + b$ with $20 - 2 a$ .

$2 = a (15) + 20 - 2 a$

$15 = 3 a + 20$

$b = 20 - 2 a$

Simplify $2 = a (15) + 20 - 2 a$ .

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Simplify the left side.

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Remove parentheses.

$2 = a (15) + 20 - 2 a$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = a (15) + 20 - 2 a$

$15 = 3 a + 20$

$b = 20 - 2 a$

Simplify the right side.

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Simplify $a (15) + 20 - 2 a$ .

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Move $15$ to the left of $a$ .

$2 = 15 a + 20 - 2 a$

$15 = 3 a + 20$

$b = 20 - 2 a$

Subtract $2 a$ from $15 a$ .

$2 = 13 a + 20$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = 13 a + 20$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = 13 a + 20$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = 13 a + 20$

$15 = 3 a + 20$

$b = 20 - 2 a$

$2 = 13 a + 20$

$15 = 3 a + 20$

$b = 20 - 2 a$

Solve for $a$ in $2 = 13 a + 20$ .

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Rewrite the equation as $13 a + 20 = 2$ .

$13 a + 20 = 2$

$15 = 3 a + 20$

$b = 20 - 2 a$

Move all terms not containing $a$ to the right side of the equation.

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Subtract $20$ from both sides of the equation.

$13 a = 2 - 20$

$15 = 3 a + 20$

$b = 20 - 2 a$

Subtract $20$ from $2$ .

$13 a = - 18$

$15 = 3 a + 20$

$b = 20 - 2 a$

$13 a = - 18$

$15 = 3 a + 20$

$b = 20 - 2 a$

Divide each term in $13 a = - 18$ by $13$ and simplify.

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Divide each term in $13 a = - 18$ by $13$ .

$\frac{13 a}{13} = \frac{- 18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

Simplify the left side.

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Cancel the common factor of $13$ .

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Cancel the common factor.

$\frac{13 a}{13} = \frac{- 18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

Divide $a$ by $1$ .

$a = \frac{- 18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

$a = \frac{- 18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

$a = \frac{- 18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

Simplify the right side.

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Move the negative in front of the fraction.

$a = - \frac{18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

$a = - \frac{18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

$a = - \frac{18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

$a = - \frac{18}{13}$

$15 = 3 a + 20$

$b = 20 - 2 a$

Replace all occurrences of $a$ with $- \frac{18}{13}$ in each equation.

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Replace all occurrences of $a$ in $15 = 3 a + 20$ with $- \frac{18}{13}$ .

$15 = 3 (- \frac{18}{13}) + 20$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Simplify the right side.

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Simplify $3 (- \frac{18}{13}) + 20$ .

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Simplify each term.

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Multiply $3 (- \frac{18}{13})$ .

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Multiply $- 1$ by $3$ .

$15 = - 3 (\frac{18}{13}) + 20$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Combine $- 3$ and $\frac{18}{13}$ .

$15 = \frac{- 3 \cdot 18}{13} + 20$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Multiply $- 3$ by $18$ .

$15 = \frac{- 54}{13} + 20$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

$15 = \frac{- 54}{13} + 20$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Move the negative in front of the fraction.

$15 = - \frac{54}{13} + 20$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

$15 = - \frac{54}{13} + 20$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

To write $20$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$15 = - \frac{54}{13} + 20 \cdot \frac{13}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Combine $20$ and $\frac{13}{13}$ .

$15 = - \frac{54}{13} + \frac{20 \cdot 13}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Combine the numerators over the common denominator.

$15 = \frac{- 54 + 20 \cdot 13}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Simplify the numerator.

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Multiply $20$ by $13$ .

$15 = \frac{- 54 + 260}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Add $- 54$ and $260$ .

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = 20 - 2 a$

Replace all occurrences of $a$ in $b = 20 - 2 a$ with $- \frac{18}{13}$ .

$b = 20 - 2 (- \frac{18}{13})$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Simplify the right side.

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Simplify $20 - 2 (- \frac{18}{13})$ .

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Multiply $- 2 (- \frac{18}{13})$ .

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Multiply $- 1$ by $- 2$ .

$b = 20 + 2 (\frac{18}{13})$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Combine $2$ and $\frac{18}{13}$ .

$b = 20 + \frac{2 \cdot 18}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Multiply $2$ by $18$ .

$b = 20 + \frac{36}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = 20 + \frac{36}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$b = 20 \cdot \frac{13}{13} + \frac{36}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Combine $20$ and $\frac{13}{13}$ .

$b = \frac{20 \cdot 13}{13} + \frac{36}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Combine the numerators over the common denominator.

$b = \frac{20 \cdot 13 + 36}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Simplify the numerator.

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Multiply $20$ by $13$ .

$b = \frac{260 + 36}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Add $260$ and $36$ .

$b = \frac{296}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = \frac{296}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = \frac{296}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = \frac{296}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

$b = \frac{296}{13}$

$15 = \frac{206}{13}$

$a = - \frac{18}{13}$

Since $15 = \frac{206}{13}$ is not true, there is no solution.

No solution

Since $y \neq y$ for the corresponding $x$ values, the function is not linear.

The function is not linear

Check if the function rule is quadratic.

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To find if the table follows a function rule, check whether the function rule could follow the form $y = a x^{2} + b x + c$ .

$y = a x^{2} + b x + c$

Build a set of $3$ equations from the table such that $y = a x^{2} + b x + c$ .

Calculate the values of $a$ , $b$ , and $c$ .

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Solve for $c$ in $20 = a \cdot 2^{2} + b (2) + c$ .

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Rewrite the equation as $a \cdot 2^{2} + b (2) + c = 20$ .

$a \cdot 2^{2} + b (2) + c = 20$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

Simplify each term.

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Raise $2$ to the power of $2$ .

$a \cdot 4 + b (2) + c = 20$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

Move $4$ to the left of $a$ .

$4 \cdot a + b (2) + c = 20$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

Move $2$ to the left of $b$ .

$4 a + 2 b + c = 20$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

$4 a + 2 b + c = 20$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

Move all terms not containing $c$ to the right side of the equation.

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Subtract $4 a$ from both sides of the equation.

$2 b + c = 20 - 4 a$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

Subtract $2 b$ from both sides of the equation.

$c = 20 - 4 a - 2 b$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

$c = 20 - 4 a - 2 b$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

$c = 20 - 4 a - 2 b$

$15 = a \cdot 5^{2} + b (5) + c$

$2 = a \cdot 15^{2} + b (15) + c$

Replace all occurrences of $c$ with $20 - 4 a - 2 b$ in each equation.

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Replace all occurrences of $c$ in $15 = a \cdot 5^{2} + b (5) + c$ with $20 - 4 a - 2 b$ .

$15 = a \cdot 5^{2} + b (5) + 20 - 4 a - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

Simplify $15 = a \cdot 5^{2} + b (5) + 20 - 4 a - 2 b$ .

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Simplify the left side.

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Remove parentheses.

$15 = a \cdot 5^{2} + b (5) + 20 - 4 a - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

$15 = a \cdot 5^{2} + b (5) + 20 - 4 a - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

Simplify the right side.

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Simplify $a \cdot 5^{2} + b (5) + 20 - 4 a - 2 b$ .

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Simplify each term.

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Raise $5$ to the power of $2$ .

$15 = a \cdot 25 + b (5) + 20 - 4 a - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

Move $25$ to the left of $a$ .

$15 = 25 \cdot a + b (5) + 20 - 4 a - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

Move $5$ to the left of $b$ .

$15 = 25 a + 5 b + 20 - 4 a - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

$15 = 25 a + 5 b + 20 - 4 a - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

Simplify by adding terms.

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Subtract $4 a$ from $25 a$ .

$15 = 21 a + 5 b + 20 - 2 b$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

Subtract $2 b$ from $5 b$ .

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + c$

Replace all occurrences of $c$ in $2 = a \cdot 15^{2} + b (15) + c$ with $20 - 4 a - 2 b$ .

$2 = a \cdot 15^{2} + b (15) + 20 - 4 a - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Simplify $2 = a \cdot 15^{2} + b (15) + 20 - 4 a - 2 b$ .

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Simplify the left side.

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Remove parentheses.

$2 = a \cdot 15^{2} + b (15) + 20 - 4 a - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = a \cdot 15^{2} + b (15) + 20 - 4 a - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Simplify the right side.

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Simplify $a \cdot 15^{2} + b (15) + 20 - 4 a - 2 b$ .

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Simplify each term.

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Raise $15$ to the power of $2$ .

$2 = a \cdot 225 + b (15) + 20 - 4 a - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Move $225$ to the left of $a$ .

$2 = 225 \cdot a + b (15) + 20 - 4 a - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Move $15$ to the left of $b$ .

$2 = 225 a + 15 b + 20 - 4 a - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = 225 a + 15 b + 20 - 4 a - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Simplify by adding terms.

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Subtract $4 a$ from $225 a$ .

$2 = 221 a + 15 b + 20 - 2 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Subtract $2 b$ from $15 b$ .

$2 = 221 a + 13 b + 20$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = 221 a + 13 b + 20$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = 221 a + 13 b + 20$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = 221 a + 13 b + 20$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = 221 a + 13 b + 20$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$2 = 221 a + 13 b + 20$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Solve for $a$ in $2 = 221 a + 13 b + 20$ .

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Rewrite the equation as $221 a + 13 b + 20 = 2$ .

$221 a + 13 b + 20 = 2$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Move all terms not containing $a$ to the right side of the equation.

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Subtract $13 b$ from both sides of the equation.

$221 a + 20 = 2 - 13 b$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Subtract $20$ from both sides of the equation.

$221 a = 2 - 13 b - 20$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Subtract $20$ from $2$ .

$221 a = - 13 b - 18$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$221 a = - 13 b - 18$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Divide each term in $221 a = - 13 b - 18$ by $221$ and simplify.

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Divide each term in $221 a = - 13 b - 18$ by $221$ .

$\frac{221 a}{221} = \frac{- 13 b}{221} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Simplify the left side.

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Cancel the common factor of $221$ .

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Cancel the common factor.

$\frac{221 a}{221} = \frac{- 13 b}{221} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Divide $a$ by $1$ .

$a = \frac{- 13 b}{221} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = \frac{- 13 b}{221} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = \frac{- 13 b}{221} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Simplify the right side.

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Simplify each term.

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Cancel the common factor of $- 13$ and $221$ .

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Factor $13$ out of $- 13 b$ .

$a = \frac{13 (- b)}{221} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Cancel the common factors.

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Factor $13$ out of $221$ .

$a = \frac{13 (- b)}{13 (17)} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Cancel the common factor.

$a = \frac{13 (- b)}{13 \cdot 17} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Rewrite the expression.

$a = \frac{- b}{17} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = \frac{- b}{17} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = \frac{- b}{17} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Move the negative in front of the fraction.

$a = - \frac{b}{17} + \frac{- 18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Move the negative in front of the fraction.

$a = - \frac{b}{17} - \frac{18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = - \frac{b}{17} - \frac{18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = - \frac{b}{17} - \frac{18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = - \frac{b}{17} - \frac{18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

$a = - \frac{b}{17} - \frac{18}{221}$

$15 = 21 a + 3 b + 20$

$c = 20 - 4 a - 2 b$

Replace all occurrences of $a$ with $- \frac{b}{17} - \frac{18}{221}$ in each equation.

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Replace all occurrences of $a$ in $15 = 21 a + 3 b + 20$ with $- \frac{b}{17} - \frac{18}{221}$ .

$15 = 21 (- \frac{b}{17} - \frac{18}{221}) + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Simplify the right side.

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Simplify $21 (- \frac{b}{17} - \frac{18}{221}) + 3 b + 20$ .

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Simplify each term.

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Apply the distributive property.

$15 = 21 (- \frac{b}{17}) + 21 (- \frac{18}{221}) + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $21 (- \frac{b}{17})$ .

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Multiply $- 1$ by $21$ .

$15 = - 21 \frac{b}{17} + 21 (- \frac{18}{221}) + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Combine $- 21$ and $\frac{b}{17}$ .

$15 = \frac{- 21 b}{17} + 21 (- \frac{18}{221}) + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{- 21 b}{17} + 21 (- \frac{18}{221}) + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $21 (- \frac{18}{221})$ .

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Multiply $- 1$ by $21$ .

$15 = \frac{- 21 b}{17} - 21 (\frac{18}{221}) + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Combine $- 21$ and $\frac{18}{221}$ .

$15 = \frac{- 21 b}{17} + \frac{- 21 \cdot 18}{221} + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $- 21$ by $18$ .

$15 = \frac{- 21 b}{17} + \frac{- 378}{221} + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{- 21 b}{17} + \frac{- 378}{221} + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Simplify each term.

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Move the negative in front of the fraction.

$15 = - \frac{(21) b}{17} + \frac{- 378}{221} + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Move the negative in front of the fraction.

$15 = - \frac{21 b}{17} - \frac{378}{221} + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = - \frac{21 b}{17} - \frac{378}{221} + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = - \frac{21 b}{17} - \frac{378}{221} + 3 b + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

To write $3 b$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$15 = - \frac{21 b}{17} + 3 b \cdot \frac{17}{17} - \frac{378}{221} + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Combine $3 b$ and $\frac{17}{17}$ .

$15 = - \frac{21 b}{17} + \frac{3 b \cdot 17}{17} - \frac{378}{221} + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Combine the numerators over the common denominator.

$15 = \frac{- 21 b + 3 b \cdot 17}{17} - \frac{378}{221} + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Find the common denominator.

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Multiply $\frac{- 21 b + 3 b \cdot 17}{17}$ by $\frac{13}{13}$ .

$15 = \frac{- 21 b + 3 b \cdot 17}{17} \cdot \frac{13}{13} - \frac{378}{221} + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $\frac{- 21 b + 3 b \cdot 17}{17}$ by $\frac{13}{13}$ .

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13}{17 \cdot 13} - \frac{378}{221} + 20$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Write $20$ as a fraction with denominator $1$ .

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13}{17 \cdot 13} - \frac{378}{221} + \frac{20}{1}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $\frac{20}{1}$ by $\frac{221}{221}$ .

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13}{17 \cdot 13} - \frac{378}{221} + \frac{20}{1} \cdot \frac{221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $\frac{20}{1}$ by $\frac{221}{221}$ .

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13}{17 \cdot 13} - \frac{378}{221} + \frac{20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Reorder the factors of $17 \cdot 13$ .

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13}{13 \cdot 17} - \frac{378}{221} + \frac{20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $13$ by $17$ .

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13}{221} - \frac{378}{221} + \frac{20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13}{221} - \frac{378}{221} + \frac{20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Combine the numerators over the common denominator.

$15 = \frac{(- 21 b + 3 b \cdot 17) \cdot 13 - 378 + 20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Simplify each term.

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Multiply $17$ by $3$ .

$15 = \frac{(- 21 b + 51 b) \cdot 13 - 378 + 20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Add $- 21 b$ and $51 b$ .

$15 = \frac{30 b \cdot 13 - 378 + 20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $13$ by $30$ .

$15 = \frac{390 b - 378 + 20 \cdot 221}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Multiply $20$ by $221$ .

$15 = \frac{390 b - 378 + 4420}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{390 b - 378 + 4420}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Simplify with factoring out.

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Add $- 378$ and $4420$ .

$15 = \frac{390 b + 4042}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Factor $2$ out of $390 b + 4042$ .

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Factor $2$ out of $390 b$ .

$15 = \frac{2 (195 b) + 4042}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Factor $2$ out of $4042$ .

$15 = \frac{2 (195 b) + 2 (2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Factor $2$ out of $2 (195 b) + 2 (2021)$ .

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 - 4 a - 2 b$

Replace all occurrences of $a$ in $c = 20 - 4 a - 2 b$ with $- \frac{b}{17} - \frac{18}{221}$ .

$c = 20 - 4 (- \frac{b}{17} - \frac{18}{221}) - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify the right side.

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Simplify $20 - 4 (- \frac{b}{17} - \frac{18}{221}) - 2 b$ .

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Simplify each term.

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Apply the distributive property.

$c = 20 - 4 (- \frac{b}{17}) - 4 (- \frac{18}{221}) - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $- 4 (- \frac{b}{17})$ .

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Multiply $- 1$ by $- 4$ .

$c = 20 + 4 (\frac{b}{17}) - 4 (- \frac{18}{221}) - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Combine $4$ and $\frac{b}{17}$ .

$c = 20 + \frac{4 b}{17} - 4 (- \frac{18}{221}) - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 + \frac{4 b}{17} - 4 (- \frac{18}{221}) - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $- 4 (- \frac{18}{221})$ .

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Multiply $- 1$ by $- 4$ .

$c = 20 + \frac{4 b}{17} + 4 (\frac{18}{221}) - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Combine $4$ and $\frac{18}{221}$ .

$c = 20 + \frac{4 b}{17} + \frac{4 \cdot 18}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $4$ by $18$ .

$c = 20 + \frac{4 b}{17} + \frac{72}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 + \frac{4 b}{17} + \frac{72}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = 20 + \frac{4 b}{17} + \frac{72}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

To write $20$ as a fraction with a common denominator, multiply by $\frac{221}{221}$ .

$c = \frac{4 b}{17} + 20 \cdot \frac{221}{221} + \frac{72}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Combine $20$ and $\frac{221}{221}$ .

$c = \frac{4 b}{17} + \frac{20 \cdot 221}{221} + \frac{72}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Combine the numerators over the common denominator.

$c = \frac{4 b}{17} + \frac{20 \cdot 221 + 72}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify the numerator.

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Multiply $20$ by $221$ .

$c = \frac{4 b}{17} + \frac{4420 + 72}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Add $4420$ and $72$ .

$c = \frac{4 b}{17} + \frac{4492}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{4 b}{17} + \frac{4492}{221} - 2 b$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

To write $- 2 b$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$c = \frac{4 b}{17} - 2 b \cdot \frac{17}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify terms.

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Combine $- 2 b$ and $\frac{17}{17}$ .

$c = \frac{4 b}{17} + \frac{- 2 b \cdot 17}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Combine the numerators over the common denominator.

$c = \frac{4 b - 2 b \cdot 17}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{4 b - 2 b \cdot 17}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify each term.

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Simplify the numerator.

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Factor $2 b$ out of $4 b - 2 b \cdot 17$ .

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Factor $2 b$ out of $4 b$ .

$c = \frac{2 b (2) - 2 b \cdot 17}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Factor $2 b$ out of $- 2 b \cdot 17$ .

$c = \frac{2 b (2) + 2 b (- 1 \cdot 17)}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Factor $2 b$ out of $2 b (2) + 2 b (- 1 \cdot 17)$ .

$c = \frac{2 b (2 - 1 \cdot 17)}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{2 b (2 - 1 \cdot 17)}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $- 1$ by $17$ .

$c = \frac{2 b (2 - 17)}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Subtract $17$ from $2$ .

$c = \frac{2 b \cdot - 15}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $- 15$ by $2$ .

$c = \frac{- 30 b}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{- 30 b}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Move the negative in front of the fraction.

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$15 = \frac{2 (195 b + 2021)}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Solve for $b$ in $15 = \frac{2 (195 b + 2021)}{221}$ .

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Rewrite the equation as $\frac{2 (195 b + 2021)}{221} = 15$ .

$\frac{2 (195 b + 2021)}{221} = 15$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply both sides by $221$ .

$\frac{2 (195 b + 2021)}{221} \cdot 221 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify.

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Simplify the left side.

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Simplify $\frac{2 (195 b + 2021)}{221} \cdot 221$ .

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Cancel the common factor of $221$ .

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Cancel the common factor.

$\frac{2 (195 b + 2021)}{221} \cdot 221 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Rewrite the expression.

$2 (195 b + 2021) = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$2 (195 b + 2021) = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Apply the distributive property.

$2 (195 b) + 2 \cdot 2021 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply.

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Multiply $195$ by $2$ .

$390 b + 2 \cdot 2021 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $2$ by $2021$ .

$390 b + 4042 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$390 b + 4042 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$390 b + 4042 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$390 b + 4042 = 15 \cdot 221$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify the right side.

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Multiply $15$ by $221$ .

$390 b + 4042 = 3315$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$390 b + 4042 = 3315$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$390 b + 4042 = 3315$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Solve for $b$ .

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Move all terms not containing $b$ to the right side of the equation.

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Subtract $4042$ from both sides of the equation.

$390 b = 3315 - 4042$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Subtract $4042$ from $3315$ .

$390 b = - 727$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$390 b = - 727$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Divide each term in $390 b = - 727$ by $390$ and simplify.

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Divide each term in $390 b = - 727$ by $390$ .

$\frac{390 b}{390} = \frac{- 727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify the left side.

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Cancel the common factor of $390$ .

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Cancel the common factor.

$\frac{390 b}{390} = \frac{- 727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Divide $b$ by $1$ .

$b = \frac{- 727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$b = \frac{- 727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$b = \frac{- 727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify the right side.

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Move the negative in front of the fraction.

$b = - \frac{727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$b = - \frac{727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$b = - \frac{727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$b = - \frac{727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

$b = - \frac{727}{390}$

$c = - \frac{30 b}{17} + \frac{4492}{221}$

$a = - \frac{b}{17} - \frac{18}{221}$

Replace all occurrences of $b$ with $- \frac{727}{390}$ in each equation.

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Replace all occurrences of $b$ in $c = - \frac{30 b}{17} + \frac{4492}{221}$ with $- \frac{727}{390}$ .

$c = - \frac{30 (- \frac{727}{390})}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify the right side.

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Simplify $- \frac{30 (- \frac{727}{390})}{17} + \frac{4492}{221}$ .

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Simplify each term.

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Simplify the numerator.

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Multiply $- 1$ by $30$ .

$c = - \frac{- 30 (\frac{727}{390})}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Combine $- 30$ and $\frac{727}{390}$ .

$c = - \frac{\frac{- 30 \cdot 727}{390}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{\frac{- 30 \cdot 727}{390}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $- 30$ by $727$ .

$c = - \frac{\frac{- 21810}{390}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Reduce the expression by cancelling the common factors.

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Cancel the common factor of $- 21810$ and $390$ .

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Factor $30$ out of $- 21810$ .

$c = - \frac{\frac{30 (- 727)}{390}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Cancel the common factors.

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Factor $30$ out of $390$ .

$c = - \frac{\frac{30 \cdot - 727}{30 \cdot 13}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Cancel the common factor.

$c = - \frac{\frac{30 \cdot - 727}{30 \cdot 13}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Rewrite the expression.

$c = - \frac{\frac{- 727}{13}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{\frac{- 727}{13}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{\frac{- 727}{13}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Move the negative in front of the fraction.

$c = - \frac{- \frac{727}{13}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = - \frac{- \frac{727}{13}}{17} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply the numerator by the reciprocal of the denominator.

$c = - (- \frac{727}{13} \cdot \frac{1}{17}) + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $- \frac{727}{13} \cdot \frac{1}{17}$ .

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Multiply $\frac{1}{17}$ by $\frac{727}{13}$ .

$c = \frac{727}{17 \cdot 13} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $17$ by $13$ .

$c = \frac{727}{221} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{727}{221} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $- - \frac{727}{221}$ .

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Multiply $- 1$ by $- 1$ .

$c = 1 (\frac{727}{221}) + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Multiply $\frac{727}{221}$ by $1$ .

$c = \frac{727}{221} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{727}{221} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{727}{221} + \frac{4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Simplify terms.

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Combine the numerators over the common denominator.

$c = \frac{727 + 4492}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Add $727$ and $4492$ .

$c = \frac{5219}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Cancel the common factor of $5219$ and $221$ .

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Factor $17$ out of $5219$ .

$c = \frac{17 (307)}{221}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Cancel the common factors.

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Factor $17$ out of $221$ .

$c = \frac{17 \cdot 307}{17 \cdot 13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Cancel the common factor.

$c = \frac{17 \cdot 307}{17 \cdot 13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Rewrite the expression.

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = - \frac{b}{17} - \frac{18}{221}$

Replace all occurrences of $b$ in $a = - \frac{b}{17} - \frac{18}{221}$ with $- \frac{727}{390}$ .

$a = - \frac{- \frac{727}{390}}{17} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Simplify the right side.

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Simplify $- \frac{- \frac{727}{390}}{17} - \frac{18}{221}$ .

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Simplify each term.

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Multiply the numerator by the reciprocal of the denominator.

$a = - (- \frac{727}{390} \cdot \frac{1}{17}) - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Multiply $- \frac{727}{390} \cdot \frac{1}{17}$ .

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Multiply $\frac{1}{17}$ by $\frac{727}{390}$ .

$a = \frac{727}{17 \cdot 390} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Multiply $17$ by $390$ .

$a = \frac{727}{6630} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{727}{6630} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Multiply $- - \frac{727}{6630}$ .

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Multiply $- 1$ by $- 1$ .

$a = 1 (\frac{727}{6630}) - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Multiply $\frac{727}{6630}$ by $1$ .

$a = \frac{727}{6630} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{727}{6630} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{727}{6630} - \frac{18}{221}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

To write $- \frac{18}{221}$ as a fraction with a common denominator, multiply by $\frac{30}{30}$ .

$a = \frac{727}{6630} - \frac{18}{221} \cdot \frac{30}{30}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Write each expression with a common denominator of $6630$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{18}{221}$ by $\frac{30}{30}$ .

$a = \frac{727}{6630} - \frac{18 \cdot 30}{221 \cdot 30}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Multiply $221$ by $30$ .

$a = \frac{727}{6630} - \frac{18 \cdot 30}{6630}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{727}{6630} - \frac{18 \cdot 30}{6630}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Combine the numerators over the common denominator.

$a = \frac{727 - 18 \cdot 30}{6630}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Simplify the numerator.

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Multiply $- 18$ by $30$ .

$a = \frac{727 - 540}{6630}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Subtract $540$ from $727$ .

$a = \frac{187}{6630}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{187}{6630}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Cancel the common factor of $187$ and $6630$ .

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Factor $17$ out of $187$ .

$a = \frac{17 (11)}{6630}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Cancel the common factors.

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Factor $17$ out of $6630$ .

$a = \frac{17 \cdot 11}{17 \cdot 390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Cancel the common factor.

$a = \frac{17 \cdot 11}{17 \cdot 390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

Rewrite the expression.

$a = \frac{11}{390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{11}{390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{11}{390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{11}{390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{11}{390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

$a = \frac{11}{390}$

$c = \frac{307}{13}$

$b = - \frac{727}{390}$

List all of the solutions.

$a = \frac{11}{390}, c = \frac{307}{13}, b = - \frac{727}{390}$

Calculate the value of $y$ using each $x$ value in the table and compare this value to the given $y$ value in the table.

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Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{11}{390}$ , $b = - \frac{727}{390}$ , $c = \frac{307}{13}$ , and $x = 2$ .

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Simplify each term.

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Raise $2$ to the power of $2$ .

$y = \frac{11}{390} \cdot 4 + (- \frac{727}{390}) \cdot (2) + \frac{307}{13}$

Cancel the common factor of $2$ .

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Factor $2$ out of $390$ .

$y = \frac{11}{2 (195)} \cdot 4 + (- \frac{727}{390}) \cdot (2) + \frac{307}{13}$

Factor $2$ out of $4$ .

$y = \frac{11}{2 \cdot 195} \cdot (2 \cdot 2) + (- \frac{727}{390}) \cdot (2) + \frac{307}{13}$

Cancel the common factor.

$y = \frac{11}{2 \cdot 195} \cdot (2 \cdot 2) + (- \frac{727}{390}) \cdot (2) + \frac{307}{13}$

Rewrite the expression.

$y = \frac{11}{195} \cdot 2 + (- \frac{727}{390}) \cdot (2) + \frac{307}{13}$

Combine $\frac{11}{195}$ and $2$ .

$y = \frac{11 \cdot 2}{195} + (- \frac{727}{390}) \cdot (2) + \frac{307}{13}$

Multiply $11$ by $2$ .

$y = \frac{22}{195} + (- \frac{727}{390}) \cdot (2) + \frac{307}{13}$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{727}{390}$ into the numerator.

$y = \frac{22}{195} + \frac{- 727}{390} \cdot 2 + \frac{307}{13}$

Factor $2$ out of $390$ .

$y = \frac{22}{195} + \frac{- 727}{2 (195)} \cdot 2 + \frac{307}{13}$

Cancel the common factor.

$y = \frac{22}{195} + \frac{- 727}{2 \cdot 195} \cdot 2 + \frac{307}{13}$

Rewrite the expression.

$y = \frac{22}{195} + \frac{- 727}{195} + \frac{307}{13}$

Move the negative in front of the fraction.

$y = \frac{22}{195} - \frac{727}{195} + \frac{307}{13}$

Combine fractions.

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Combine the numerators over the common denominator.

$y = \frac{22 - 727}{195} + \frac{307}{13}$

Subtract $727$ from $22$ .

$y = \frac{- 705}{195} + \frac{307}{13}$

Simplify each term.

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Cancel the common factor of $- 705$ and $195$ .

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Factor $15$ out of $- 705$ .

$y = \frac{15 (- 47)}{195} + \frac{307}{13}$

Cancel the common factors.

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Factor $15$ out of $195$ .

$y = \frac{15 \cdot - 47}{15 \cdot 13} + \frac{307}{13}$

Cancel the common factor.

$y = \frac{15 \cdot - 47}{15 \cdot 13} + \frac{307}{13}$

Rewrite the expression.

$y = \frac{- 47}{13} + \frac{307}{13}$

Move the negative in front of the fraction.

$y = - \frac{47}{13} + \frac{307}{13}$

Combine fractions.

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Combine the numerators over the common denominator.

$y = \frac{- 47 + 307}{13}$

Simplify the expression.

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Add $- 47$ and $307$ .

$y = \frac{260}{13}$

Divide $260$ by $13$ .

$y = 20$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 20$ and $y = 20$ .

$20 = 20$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{11}{390}$ , $b = - \frac{727}{390}$ , $c = \frac{307}{13}$ , and $x = 5$ .

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Simplify each term.

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Raise $5$ to the power of $2$ .

$y = \frac{11}{390} \cdot 25 + (- \frac{727}{390}) \cdot (5) + \frac{307}{13}$

Cancel the common factor of $5$ .

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Factor $5$ out of $390$ .

$y = \frac{11}{5 (78)} \cdot 25 + (- \frac{727}{390}) \cdot (5) + \frac{307}{13}$

Factor $5$ out of $25$ .

$y = \frac{11}{5 \cdot 78} \cdot (5 \cdot 5) + (- \frac{727}{390}) \cdot (5) + \frac{307}{13}$

Cancel the common factor.

$y = \frac{11}{5 \cdot 78} \cdot (5 \cdot 5) + (- \frac{727}{390}) \cdot (5) + \frac{307}{13}$

Rewrite the expression.

$y = \frac{11}{78} \cdot 5 + (- \frac{727}{390}) \cdot (5) + \frac{307}{13}$

Combine $\frac{11}{78}$ and $5$ .

$y = \frac{11 \cdot 5}{78} + (- \frac{727}{390}) \cdot (5) + \frac{307}{13}$

Multiply $11$ by $5$ .

$y = \frac{55}{78} + (- \frac{727}{390}) \cdot (5) + \frac{307}{13}$

Cancel the common factor of $5$ .

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Move the leading negative in $- \frac{727}{390}$ into the numerator.

$y = \frac{55}{78} + \frac{- 727}{390} \cdot 5 + \frac{307}{13}$

Factor $5$ out of $390$ .

$y = \frac{55}{78} + \frac{- 727}{5 (78)} \cdot 5 + \frac{307}{13}$

Cancel the common factor.

$y = \frac{55}{78} + \frac{- 727}{5 \cdot 78} \cdot 5 + \frac{307}{13}$

Rewrite the expression.

$y = \frac{55}{78} + \frac{- 727}{78} + \frac{307}{13}$

Move the negative in front of the fraction.

$y = \frac{55}{78} - \frac{727}{78} + \frac{307}{13}$

Combine fractions.

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Combine the numerators over the common denominator.

$y = \frac{55 - 727}{78} + \frac{307}{13}$

Subtract $727$ from $55$ .

$y = \frac{- 672}{78} + \frac{307}{13}$

Simplify each term.

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Cancel the common factor of $- 672$ and $78$ .

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Factor $6$ out of $- 672$ .

$y = \frac{6 (- 112)}{78} + \frac{307}{13}$

Cancel the common factors.

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Factor $6$ out of $78$ .

$y = \frac{6 \cdot - 112}{6 \cdot 13} + \frac{307}{13}$

Cancel the common factor.

$y = \frac{6 \cdot - 112}{6 \cdot 13} + \frac{307}{13}$

Rewrite the expression.

$y = \frac{- 112}{13} + \frac{307}{13}$

Move the negative in front of the fraction.

$y = - \frac{112}{13} + \frac{307}{13}$

Combine fractions.

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Combine the numerators over the common denominator.

$y = \frac{- 112 + 307}{13}$

Simplify the expression.

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Add $- 112$ and $307$ .

$y = \frac{195}{13}$

Divide $195$ by $13$ .

$y = 15$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 5$ . This check passes since $y = 15$ and $y = 15$ .

$15 = 15$

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{11}{390}$ , $b = - \frac{727}{390}$ , $c = \frac{307}{13}$ , and $x = 15$ .

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Simplify each term.

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Raise $15$ to the power of $2$ .

$y = \frac{11}{390} \cdot 225 + (- \frac{727}{390}) \cdot (15) + \frac{307}{13}$

Cancel the common factor of $15$ .

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Factor $15$ out of $390$ .

$y = \frac{11}{15 (26)} \cdot 225 + (- \frac{727}{390}) \cdot (15) + \frac{307}{13}$

Factor $15$ out of $225$ .

$y = \frac{11}{15 \cdot 26} \cdot (15 \cdot 15) + (- \frac{727}{390}) \cdot (15) + \frac{307}{13}$

Cancel the common factor.

$y = \frac{11}{15 \cdot 26} \cdot (15 \cdot 15) + (- \frac{727}{390}) \cdot (15) + \frac{307}{13}$

Rewrite the expression.

$y = \frac{11}{26} \cdot 15 + (- \frac{727}{390}) \cdot (15) + \frac{307}{13}$

Combine $\frac{11}{26}$ and $15$ .

$y = \frac{11 \cdot 15}{26} + (- \frac{727}{390}) \cdot (15) + \frac{307}{13}$

Multiply $11$ by $15$ .

$y = \frac{165}{26} + (- \frac{727}{390}) \cdot (15) + \frac{307}{13}$

Cancel the common factor of $15$ .

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Move the leading negative in $- \frac{727}{390}$ into the numerator.

$y = \frac{165}{26} + \frac{- 727}{390} \cdot 15 + \frac{307}{13}$

Factor $15$ out of $390$ .

$y = \frac{165}{26} + \frac{- 727}{15 (26)} \cdot 15 + \frac{307}{13}$

Cancel the common factor.

$y = \frac{165}{26} + \frac{- 727}{15 \cdot 26} \cdot 15 + \frac{307}{13}$

Rewrite the expression.

$y = \frac{165}{26} + \frac{- 727}{26} + \frac{307}{13}$

Move the negative in front of the fraction.

$y = \frac{165}{26} - \frac{727}{26} + \frac{307}{13}$

Combine fractions.

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Combine the numerators over the common denominator.

$y = \frac{165 - 727}{26} + \frac{307}{13}$

Subtract $727$ from $165$ .

$y = \frac{- 562}{26} + \frac{307}{13}$

Simplify each term.

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Cancel the common factor of $- 562$ and $26$ .

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Factor $2$ out of $- 562$ .

$y = \frac{2 (- 281)}{26} + \frac{307}{13}$

Cancel the common factors.

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Factor $2$ out of $26$ .

$y = \frac{2 \cdot - 281}{2 \cdot 13} + \frac{307}{13}$

Cancel the common factor.

$y = \frac{2 \cdot - 281}{2 \cdot 13} + \frac{307}{13}$

Rewrite the expression.

$y = \frac{- 281}{13} + \frac{307}{13}$

Move the negative in front of the fraction.

$y = - \frac{281}{13} + \frac{307}{13}$

Combine fractions.

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Combine the numerators over the common denominator.

$y = \frac{- 281 + 307}{13}$

Simplify the expression.

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Add $- 281$ and $307$ .

$y = \frac{26}{13}$

Divide $26$ by $13$ .

$y = 2$

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 15$ . This check passes since $y = 2$ and $y = 2$ .

$2 = 2$

Since $y = y$ for the corresponding $x$ values, the function is quadratic.

The function is quadratic

Since all $y = y$ , the function is quadratic and follows the form $y = \frac{11 x^{2}}{390} - \frac{727 x}{390} + \frac{307}{13}$ .

$y = \frac{11 x^{2}}{390} - \frac{727 x}{390} + \frac{307}{13}$