Algebra Examples

$\begin{array}{c} x & y \\ 4 & 4 \\ 8 & 16 \\ 12 & 64 \end{array}$

Step 1

Check if the function rule is linear.

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Step 1.1

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

$y = a x + b$

Step 1.2

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

$4 = a (4) + b 16 = a (8) + b 64 = a (12) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $4 = a (4) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (4) + b = 4$ .

$a (4) + b = 4$

$16 = a (8) + b$

$64 = a (12) + b$

Step 1.3.1.2

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

$4 a + b = 4$

$16 = a (8) + b$

$64 = a (12) + b$

Step 1.3.1.3

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

$b = 4 - 4 a$

$16 = a (8) + b$

$64 = a (12) + b$

$b = 4 - 4 a$

$16 = a (8) + b$

$64 = a (12) + b$

Step 1.3.2

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

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Step 1.3.2.1

Replace all occurrences of $b$ in $16 = a (8) + b$ with $4 - 4 a$ .

$16 = a (8) + 4 - 4 a$

$b = 4 - 4 a$

$64 = a (12) + b$

Step 1.3.2.2

Simplify $16 = a (8) + 4 - 4 a$ .

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Step 1.3.2.2.1

Simplify the left side.

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Step 1.3.2.2.1.1

Remove parentheses.

$16 = a (8) + 4 - 4 a$

$b = 4 - 4 a$

$64 = a (12) + b$

$16 = a (8) + 4 - 4 a$

$b = 4 - 4 a$

$64 = a (12) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (8) + 4 - 4 a$ .

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Step 1.3.2.2.2.1.1

Move $8$ to the left of $a$ .

$16 = 8 a + 4 - 4 a$

$b = 4 - 4 a$

$64 = a (12) + b$

Step 1.3.2.2.2.1.2

Subtract $4 a$ from $8 a$ .

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = a (12) + b$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = a (12) + b$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = a (12) + b$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = a (12) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $64 = a (12) + b$ with $4 - 4 a$ .

$64 = a (12) + 4 - 4 a$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.2.4

Simplify $64 = a (12) + 4 - 4 a$ .

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Step 1.3.2.4.1

Simplify the left side.

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Step 1.3.2.4.1.1

Remove parentheses.

$64 = a (12) + 4 - 4 a$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = a (12) + 4 - 4 a$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (12) + 4 - 4 a$ .

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Step 1.3.2.4.2.1.1

Move $12$ to the left of $a$ .

$64 = 12 a + 4 - 4 a$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.2.4.2.1.2

Subtract $4 a$ from $12 a$ .

$64 = 8 a + 4$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = 8 a + 4$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = 8 a + 4$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = 8 a + 4$

$16 = 4 a + 4$

$b = 4 - 4 a$

$64 = 8 a + 4$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3

Solve for $a$ in $64 = 8 a + 4$ .

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Step 1.3.3.1

Rewrite the equation as $8 a + 4 = 64$ .

$8 a + 4 = 64$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Subtract $4$ from both sides of the equation.

$8 a = 64 - 4$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.2.2

Subtract $4$ from $64$ .

$8 a = 60$

$16 = 4 a + 4$

$b = 4 - 4 a$

$8 a = 60$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.3

Divide each term in $8 a = 60$ by $8$ and simplify.

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Step 1.3.3.3.1

Divide each term in $8 a = 60$ by $8$ .

$\frac{8 a}{8} = \frac{60}{8}$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $8$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{8 a}{8} = \frac{60}{8}$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{60}{8}$

$16 = 4 a + 4$

$b = 4 - 4 a$

$a = \frac{60}{8}$

$16 = 4 a + 4$

$b = 4 - 4 a$

$a = \frac{60}{8}$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Cancel the common factor of $60$ and $8$ .

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Step 1.3.3.3.3.1.1

Factor $4$ out of $60$ .

$a = \frac{4 (15)}{8}$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.3.3.1.2

Cancel the common factors.

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Step 1.3.3.3.3.1.2.1

Factor $4$ out of $8$ .

$a = \frac{4 \cdot 15}{4 \cdot 2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.3.3.1.2.2

Cancel the common factor.

$a = \frac{4 \cdot 15}{4 \cdot 2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.3.3.3.1.2.3

Rewrite the expression.

$a = \frac{15}{2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

$a = \frac{15}{2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

$a = \frac{15}{2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

$a = \frac{15}{2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

$a = \frac{15}{2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

$a = \frac{15}{2}$

$16 = 4 a + 4$

$b = 4 - 4 a$

Step 1.3.4

Replace all occurrences of $a$ with $\frac{15}{2}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $16 = 4 a + 4$ with $\frac{15}{2}$ .

$16 = 4 (\frac{15}{2}) + 4$

$a = \frac{15}{2}$

$b = 4 - 4 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $4 (\frac{15}{2}) + 4$ .

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Step 1.3.4.2.1.1

Simplify each term.

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Step 1.3.4.2.1.1.1

Cancel the common factor of $2$ .

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Step 1.3.4.2.1.1.1.1

Factor $2$ out of $4$ .

$16 = 2 (2) (\frac{15}{2}) + 4$

$a = \frac{15}{2}$

$b = 4 - 4 a$

Step 1.3.4.2.1.1.1.2

Cancel the common factor.

$16 = 2 \cdot (2 (\frac{15}{2})) + 4$

$a = \frac{15}{2}$

$b = 4 - 4 a$

Step 1.3.4.2.1.1.1.3

Rewrite the expression.

$16 = 2 \cdot 15 + 4$

$a = \frac{15}{2}$

$b = 4 - 4 a$

$16 = 2 \cdot 15 + 4$

$a = \frac{15}{2}$

$b = 4 - 4 a$

Step 1.3.4.2.1.1.2

Multiply $2$ by $15$ .

$16 = 30 + 4$

$a = \frac{15}{2}$

$b = 4 - 4 a$

$16 = 30 + 4$

$a = \frac{15}{2}$

$b = 4 - 4 a$

Step 1.3.4.2.1.2

Add $30$ and $4$ .

$16 = 34$

$a = \frac{15}{2}$

$b = 4 - 4 a$

$16 = 34$

$a = \frac{15}{2}$

$b = 4 - 4 a$

$16 = 34$

$a = \frac{15}{2}$

$b = 4 - 4 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 4 - 4 a$ with $\frac{15}{2}$ .

$b = 4 - 4 (\frac{15}{2})$

$16 = 34$

$a = \frac{15}{2}$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $4 - 4 (\frac{15}{2})$ .

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Step 1.3.4.4.1.1

Simplify each term.

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Step 1.3.4.4.1.1.1

Cancel the common factor of $2$ .

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Step 1.3.4.4.1.1.1.1

Factor $2$ out of $- 4$ .

$b = 4 + 2 (- 2) (\frac{15}{2})$

$16 = 34$

$a = \frac{15}{2}$

Step 1.3.4.4.1.1.1.2

Cancel the common factor.

$b = 4 + 2 \cdot (- 2 (\frac{15}{2}))$

$16 = 34$

$a = \frac{15}{2}$

Step 1.3.4.4.1.1.1.3

Rewrite the expression.

$b = 4 - 2 \cdot 15$

$16 = 34$

$a = \frac{15}{2}$

$b = 4 - 2 \cdot 15$

$16 = 34$

$a = \frac{15}{2}$

Step 1.3.4.4.1.1.2

Multiply $- 2$ by $15$ .

$b = 4 - 30$

$16 = 34$

$a = \frac{15}{2}$

$b = 4 - 30$

$16 = 34$

$a = \frac{15}{2}$

Step 1.3.4.4.1.2

Subtract $30$ from $4$ .

$b = - 26$

$16 = 34$

$a = \frac{15}{2}$

$b = - 26$

$16 = 34$

$a = \frac{15}{2}$

$b = - 26$

$16 = 34$

$a = \frac{15}{2}$

$b = - 26$

$16 = 34$

$a = \frac{15}{2}$

Step 1.3.5

Since $16 = 34$ is not true, there is no solution.

No solution

Step 1.4

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

The function is not linear

Step 2

Check if the function rule is quadratic.

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Step 2.1

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

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

Step 2.2

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

Step 2.3

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

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Step 2.3.1

Solve for $c$ in $4 = a \cdot 4^{2} + b (4) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a \cdot 4^{2} + b (4) + c = 4$ .

$a \cdot 4^{2} + b (4) + c = 4$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

Raise $4$ to the power of $2$ .

$a \cdot 16 + b (4) + c = 4$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.2.2

Move $16$ to the left of $a$ .

$16 \cdot a + b (4) + c = 4$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.2.3

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

$16 a + 4 b + c = 4$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

$16 a + 4 b + c = 4$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.3

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

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Step 2.3.1.3.1

Subtract $16 a$ from both sides of the equation.

$4 b + c = 4 - 16 a$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.1.3.2

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

$c = 4 - 16 a - 4 b$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

$c = 4 - 16 a - 4 b$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

$c = 4 - 16 a - 4 b$

$16 = a \cdot 8^{2} + b (8) + c$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2

Replace all occurrences of $c$ with $4 - 16 a - 4 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $16 = a \cdot 8^{2} + b (8) + c$ with $4 - 16 a - 4 b$ .

$16 = a \cdot 8^{2} + b (8) + 4 - 16 a - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2

Simplify $16 = a \cdot 8^{2} + b (8) + 4 - 16 a - 4 b$ .

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Step 2.3.2.2.1

Simplify the left side.

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Step 2.3.2.2.1.1

Remove parentheses.

$16 = a \cdot 8^{2} + b (8) + 4 - 16 a - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

$16 = a \cdot 8^{2} + b (8) + 4 - 16 a - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a \cdot 8^{2} + b (8) + 4 - 16 a - 4 b$ .

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Step 2.3.2.2.2.1.1

Simplify each term.

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Step 2.3.2.2.2.1.1.1

Raise $8$ to the power of $2$ .

$16 = a \cdot 64 + b (8) + 4 - 16 a - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2.1.1.2

Move $64$ to the left of $a$ .

$16 = 64 \cdot a + b (8) + 4 - 16 a - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2.1.1.3

Move $8$ to the left of $b$ .

$16 = 64 a + 8 b + 4 - 16 a - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

$16 = 64 a + 8 b + 4 - 16 a - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.2.1.2.1

Subtract $16 a$ from $64 a$ .

$16 = 48 a + 8 b + 4 - 4 b$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.2.2.1.2.2

Subtract $4 b$ from $8 b$ .

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $64 = a \cdot 12^{2} + b (12) + c$ with $4 - 16 a - 4 b$ .

$64 = a \cdot 12^{2} + b (12) + 4 - 16 a - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.2.4

Simplify $64 = a \cdot 12^{2} + b (12) + 4 - 16 a - 4 b$ .

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Step 2.3.2.4.1

Simplify the left side.

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Step 2.3.2.4.1.1

Remove parentheses.

$64 = a \cdot 12^{2} + b (12) + 4 - 16 a - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = a \cdot 12^{2} + b (12) + 4 - 16 a - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a \cdot 12^{2} + b (12) + 4 - 16 a - 4 b$ .

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Step 2.3.2.4.2.1.1

Simplify each term.

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Step 2.3.2.4.2.1.1.1

Raise $12$ to the power of $2$ .

$64 = a \cdot 144 + b (12) + 4 - 16 a - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.2.4.2.1.1.2

Move $144$ to the left of $a$ .

$64 = 144 \cdot a + b (12) + 4 - 16 a - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.2.4.2.1.1.3

Move $12$ to the left of $b$ .

$64 = 144 a + 12 b + 4 - 16 a - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = 144 a + 12 b + 4 - 16 a - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.2.4.2.1.2

Simplify by adding terms.

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Step 2.3.2.4.2.1.2.1

Subtract $16 a$ from $144 a$ .

$64 = 128 a + 12 b + 4 - 4 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.2.4.2.1.2.2

Subtract $4 b$ from $12 b$ .

$64 = 128 a + 8 b + 4$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = 128 a + 8 b + 4$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = 128 a + 8 b + 4$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = 128 a + 8 b + 4$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = 128 a + 8 b + 4$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$64 = 128 a + 8 b + 4$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3

Solve for $a$ in $64 = 128 a + 8 b + 4$ .

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Step 2.3.3.1

Rewrite the equation as $128 a + 8 b + 4 = 64$ .

$128 a + 8 b + 4 = 64$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Subtract $8 b$ from both sides of the equation.

$128 a + 4 = 64 - 8 b$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.2.2

Subtract $4$ from both sides of the equation.

$128 a = 64 - 8 b - 4$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.2.3

Subtract $4$ from $64$ .

$128 a = - 8 b + 60$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$128 a = - 8 b + 60$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3

Divide each term in $128 a = - 8 b + 60$ by $128$ and simplify.

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Step 2.3.3.3.1

Divide each term in $128 a = - 8 b + 60$ by $128$ .

$\frac{128 a}{128} = \frac{- 8 b}{128} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.2

Simplify the left side.

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Step 2.3.3.3.2.1

Cancel the common factor of $128$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{128 a}{128} = \frac{- 8 b}{128} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 8 b}{128} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = \frac{- 8 b}{128} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = \frac{- 8 b}{128} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3

Simplify the right side.

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Step 2.3.3.3.3.1

Simplify each term.

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Step 2.3.3.3.3.1.1

Cancel the common factor of $- 8$ and $128$ .

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Step 2.3.3.3.3.1.1.1

Factor $8$ out of $- 8 b$ .

$a = \frac{8 (- b)}{128} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.1.2

Cancel the common factors.

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Step 2.3.3.3.3.1.1.2.1

Factor $8$ out of $128$ .

$a = \frac{8 (- b)}{8 (16)} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{8 (- b)}{8 \cdot 16} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{16} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = \frac{- b}{16} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = \frac{- b}{16} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{16} + \frac{60}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.3

Cancel the common factor of $60$ and $128$ .

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Step 2.3.3.3.3.1.3.1

Factor $4$ out of $60$ .

$a = - \frac{b}{16} + \frac{4 (15)}{128}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.3.2

Cancel the common factors.

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Step 2.3.3.3.3.1.3.2.1

Factor $4$ out of $128$ .

$a = - \frac{b}{16} + \frac{4 \cdot 15}{4 \cdot 32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{16} + \frac{4 \cdot 15}{4 \cdot 32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{16} + \frac{15}{32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = - \frac{b}{16} + \frac{15}{32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = - \frac{b}{16} + \frac{15}{32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = - \frac{b}{16} + \frac{15}{32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = - \frac{b}{16} + \frac{15}{32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = - \frac{b}{16} + \frac{15}{32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

$a = - \frac{b}{16} + \frac{15}{32}$

$16 = 48 a + 4 b + 4$

$c = 4 - 16 a - 4 b$

Step 2.3.4

Replace all occurrences of $a$ with $- \frac{b}{16} + \frac{15}{32}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $a$ in $16 = 48 a + 4 b + 4$ with $- \frac{b}{16} + \frac{15}{32}$ .

$16 = 48 (- \frac{b}{16} + \frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $48 (- \frac{b}{16} + \frac{15}{32}) + 4 b + 4$ .

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Step 2.3.4.2.1.1

Simplify each term.

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Step 2.3.4.2.1.1.1

Apply the distributive property.

$16 = 48 (- \frac{b}{16}) + 48 (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.2

Cancel the common factor of $16$ .

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Step 2.3.4.2.1.1.2.1

Move the leading negative in $- \frac{b}{16}$ into the numerator.

$16 = 48 (\frac{- b}{16}) + 48 (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.2.2

Factor $16$ out of $48$ .

$16 = 16 (3) (\frac{- b}{16}) + 48 (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.2.3

Cancel the common factor.

$16 = 16 \cdot (3 (\frac{- b}{16})) + 48 (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.2.4

Rewrite the expression.

$16 = 3 (- b) + 48 (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

$16 = 3 (- b) + 48 (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.3

Multiply $- 1$ by $3$ .

$16 = - 3 b + 48 (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.4

Cancel the common factor of $16$ .

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Step 2.3.4.2.1.1.4.1

Factor $16$ out of $48$ .

$16 = - 3 b + 16 (3) (\frac{15}{32}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.4.2

Factor $16$ out of $32$ .

$16 = - 3 b + 16 \cdot (3 (\frac{15}{16 \cdot 2})) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.4.3

Cancel the common factor.

$16 = - 3 b + 16 \cdot (3 (\frac{15}{16 \cdot 2})) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.4.4

Rewrite the expression.

$16 = - 3 b + 3 (\frac{15}{2}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

$16 = - 3 b + 3 (\frac{15}{2}) + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.5

Combine $3$ and $\frac{15}{2}$ .

$16 = - 3 b + \frac{3 \cdot 15}{2} + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.1.6

Multiply $3$ by $15$ .

$16 = - 3 b + \frac{45}{2} + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

$16 = - 3 b + \frac{45}{2} + 4 b + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.2

Add $- 3 b$ and $4 b$ .

$16 = b + \frac{45}{2} + 4$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.3

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$16 = b + \frac{45}{2} + 4 \cdot \frac{2}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.4

Combine $4$ and $\frac{2}{2}$ .

$16 = b + \frac{45}{2} + \frac{4 \cdot 2}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.5

Combine the numerators over the common denominator.

$16 = b + \frac{45 + 4 \cdot 2}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.6

Simplify the numerator.

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Step 2.3.4.2.1.6.1

Multiply $4$ by $2$ .

$16 = b + \frac{45 + 8}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.2.1.6.2

Add $45$ and $8$ .

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 16 a - 4 b$

Step 2.3.4.3

Replace all occurrences of $a$ in $c = 4 - 16 a - 4 b$ with $- \frac{b}{16} + \frac{15}{32}$ .

$c = 4 - 16 (- \frac{b}{16} + \frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $4 - 16 (- \frac{b}{16} + \frac{15}{32}) - 4 b$ .

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Step 2.3.4.4.1.1

Simplify each term.

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Step 2.3.4.4.1.1.1

Apply the distributive property.

$c = 4 - 16 (- \frac{b}{16}) - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.2

Cancel the common factor of $16$ .

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Step 2.3.4.4.1.1.2.1

Move the leading negative in $- \frac{b}{16}$ into the numerator.

$c = 4 - 16 \frac{- b}{16} - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.2.2

Factor $16$ out of $- 16$ .

$c = 4 + 16 (- 1) (\frac{- b}{16}) - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.2.3

Cancel the common factor.

$c = 4 + 16 \cdot (- 1 \frac{- b}{16}) - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.2.4

Rewrite the expression.

$c = 4 - 1 (- b) - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 - 1 (- b) - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.3

Multiply $- 1$ by $- 1$ .

$c = 4 + 1 b - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.4

Multiply $b$ by $1$ .

$c = 4 + b - 16 (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.5

Cancel the common factor of $16$ .

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Step 2.3.4.4.1.1.5.1

Factor $16$ out of $- 16$ .

$c = 4 + b + 16 (- 1) (\frac{15}{32}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.5.2

Factor $16$ out of $32$ .

$c = 4 + b + 16 \cdot (- 1 \frac{15}{16 \cdot 2}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.5.3

Cancel the common factor.

$c = 4 + b + 16 \cdot (- 1 \frac{15}{16 \cdot 2}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.5.4

Rewrite the expression.

$c = 4 + b - 1 (\frac{15}{2}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 + b - 1 (\frac{15}{2}) - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.1.6

Rewrite $- 1 (\frac{15}{2})$ as $- (\frac{15}{2})$ .

$c = 4 + b - \frac{15}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 4 + b - \frac{15}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$c = b + 4 \cdot \frac{2}{2} - \frac{15}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.3

Combine $4$ and $\frac{2}{2}$ .

$c = b + \frac{4 \cdot 2}{2} - \frac{15}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$c = b + \frac{4 \cdot 2 - 15}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $4$ by $2$ .

$c = b + \frac{8 - 15}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.5.2

Subtract $15$ from $8$ .

$c = b + \frac{- 7}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = b + \frac{- 7}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.6

Move the negative in front of the fraction.

$c = b - \frac{7}{2} - 4 b$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.4.4.1.7

Subtract $4 b$ from $b$ .

$c = - 3 b - \frac{7}{2}$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = - 3 b - \frac{7}{2}$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = - 3 b - \frac{7}{2}$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = - 3 b - \frac{7}{2}$

$16 = b + \frac{53}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5

Solve for $b$ in $16 = b + \frac{53}{2}$ .

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Step 2.3.5.1

Rewrite the equation as $b + \frac{53}{2} = 16$ .

$b + \frac{53}{2} = 16$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Subtract $\frac{53}{2}$ from both sides of the equation.

$b = 16 - \frac{53}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5.2.2

To write $16$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = 16 \cdot \frac{2}{2} - \frac{53}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5.2.3

Combine $16$ and $\frac{2}{2}$ .

$b = \frac{16 \cdot 2}{2} - \frac{53}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5.2.4

Combine the numerators over the common denominator.

$b = \frac{16 \cdot 2 - 53}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5.2.5

Simplify the numerator.

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Step 2.3.5.2.5.1

Multiply $16$ by $2$ .

$b = \frac{32 - 53}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5.2.5.2

Subtract $53$ from $32$ .

$b = \frac{- 21}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$b = \frac{- 21}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.5.2.6

Move the negative in front of the fraction.

$b = - \frac{21}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$b = - \frac{21}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$b = - \frac{21}{2}$

$c = - 3 b - \frac{7}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6

Replace all occurrences of $b$ with $- \frac{21}{2}$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $b$ in $c = - 3 b - \frac{7}{2}$ with $- \frac{21}{2}$ .

$c = - 3 (- \frac{21}{2}) - \frac{7}{2}$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- 3 (- \frac{21}{2}) - \frac{7}{2}$ .

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Step 2.3.6.2.1.1

Multiply $- 3 (- \frac{21}{2})$ .

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Step 2.3.6.2.1.1.1

Multiply $- 1$ by $- 3$ .

$c = 3 (\frac{21}{2}) - \frac{7}{2}$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6.2.1.1.2

Combine $3$ and $\frac{21}{2}$ .

$c = \frac{3 \cdot 21}{2} - \frac{7}{2}$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6.2.1.1.3

Multiply $3$ by $21$ .

$c = \frac{63}{2} - \frac{7}{2}$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = \frac{63}{2} - \frac{7}{2}$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6.2.1.2

Combine the numerators over the common denominator.

$c = \frac{63 - 7}{2}$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6.2.1.3

Simplify the expression.

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Step 2.3.6.2.1.3.1

Subtract $7$ from $63$ .

$c = \frac{56}{2}$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6.2.1.3.2

Divide $56$ by $2$ .

$c = 28$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

$a = - \frac{b}{16} + \frac{15}{32}$

Step 2.3.6.3

Replace all occurrences of $b$ in $a = - \frac{b}{16} + \frac{15}{32}$ with $- \frac{21}{2}$ .

$a = - \frac{- \frac{21}{2}}{16} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $- \frac{- \frac{21}{2}}{16} + \frac{15}{32}$ .

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Step 2.3.6.4.1.1

Simplify each term.

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Step 2.3.6.4.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - (- \frac{21}{2} \cdot \frac{1}{16}) + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.1.2

Multiply $- \frac{21}{2} \cdot \frac{1}{16}$ .

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Step 2.3.6.4.1.1.2.1

Multiply $\frac{1}{16}$ by $\frac{21}{2}$ .

$a = \frac{21}{16 \cdot 2} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.1.2.2

Multiply $16$ by $2$ .

$a = \frac{21}{32} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{21}{32} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.1.3

Multiply $- - \frac{21}{32}$ .

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Step 2.3.6.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$a = 1 (\frac{21}{32}) + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.1.3.2

Multiply $\frac{21}{32}$ by $1$ .

$a = \frac{21}{32} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{21}{32} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{21}{32} + \frac{15}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.2

Simplify terms.

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Step 2.3.6.4.1.2.1

Combine the numerators over the common denominator.

$a = \frac{21 + 15}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.2.2

Add $21$ and $15$ .

$a = \frac{36}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.2.3

Cancel the common factor of $36$ and $32$ .

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Step 2.3.6.4.1.2.3.1

Factor $4$ out of $36$ .

$a = \frac{4 (9)}{32}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.2.3.2

Cancel the common factors.

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Step 2.3.6.4.1.2.3.2.1

Factor $4$ out of $32$ .

$a = \frac{4 \cdot 9}{4 \cdot 8}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.2.3.2.2

Cancel the common factor.

$a = \frac{4 \cdot 9}{4 \cdot 8}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.6.4.1.2.3.2.3

Rewrite the expression.

$a = \frac{9}{8}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{9}{8}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{9}{8}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{9}{8}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{9}{8}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{9}{8}$

$c = 28$

$b = - \frac{21}{2}$

$a = \frac{9}{8}$

$c = 28$

$b = - \frac{21}{2}$

Step 2.3.7

List all of the solutions.

$a = \frac{9}{8}, c = 28, b = - \frac{21}{2}$

Step 2.4

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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Step 2.4.1

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{9}{8}$ , $b = - \frac{21}{2}$ , $c = 28$ , and $x = 4$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

Raise $4$ to the power of $2$ .

$y = \frac{9}{8} \cdot 16 + (- \frac{21}{2}) \cdot (4) + 28$

Step 2.4.1.1.2

Cancel the common factor of $8$ .

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Step 2.4.1.1.2.1

Factor $8$ out of $16$ .

$y = \frac{9}{8} \cdot (8 (2)) + (- \frac{21}{2}) \cdot (4) + 28$

Step 2.4.1.1.2.2

Cancel the common factor.

$y = \frac{9}{8} \cdot (8 \cdot 2) + (- \frac{21}{2}) \cdot (4) + 28$

Step 2.4.1.1.2.3

Rewrite the expression.

$y = 9 \cdot 2 + (- \frac{21}{2}) \cdot (4) + 28$

Step 2.4.1.1.3

Multiply $9$ by $2$ .

$y = 18 + (- \frac{21}{2}) \cdot (4) + 28$

Step 2.4.1.1.4

Cancel the common factor of $2$ .

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Step 2.4.1.1.4.1

Move the leading negative in $- \frac{21}{2}$ into the numerator.

$y = 18 + \frac{- 21}{2} \cdot 4 + 28$

Step 2.4.1.1.4.2

Factor $2$ out of $4$ .

$y = 18 + \frac{- 21}{2} \cdot (2 (2)) + 28$

Step 2.4.1.1.4.3

Cancel the common factor.

$y = 18 + \frac{- 21}{2} \cdot (2 \cdot 2) + 28$

Step 2.4.1.1.4.4

Rewrite the expression.

$y = 18 - 21 \cdot 2 + 28$

Step 2.4.1.1.5

Multiply $- 21$ by $2$ .

$y = 18 - 42 + 28$

Step 2.4.1.2

Simplify by adding and subtracting.

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Step 2.4.1.2.1

Subtract $42$ from $18$ .

$y = - 24 + 28$

Step 2.4.1.2.2

Add $- 24$ and $28$ .

$y = 4$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = 4$ and $y = 4$ .

$4 = 4$

Step 2.4.3

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{9}{8}$ , $b = - \frac{21}{2}$ , $c = 28$ , and $x = 8$ .

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Step 2.4.3.1

Simplify each term.

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Step 2.4.3.1.1

Cancel the common factor of $8$ .

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Step 2.4.3.1.1.1

Factor $8$ out of ${(8)}^{2}$ .

$y = \frac{9}{8} \cdot (8 \cdot 8) + (- \frac{21}{2}) \cdot (8) + 28$

Step 2.4.3.1.1.2

Cancel the common factor.

$y = \frac{9}{8} \cdot (8 \cdot 8) + (- \frac{21}{2}) \cdot (8) + 28$

Step 2.4.3.1.1.3

Rewrite the expression.

$y = 9 \cdot 8 + (- \frac{21}{2}) \cdot (8) + 28$

Step 2.4.3.1.2

Multiply $9$ by $8$ .

$y = 72 + (- \frac{21}{2}) \cdot (8) + 28$

Step 2.4.3.1.3

Cancel the common factor of $2$ .

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Step 2.4.3.1.3.1

Move the leading negative in $- \frac{21}{2}$ into the numerator.

$y = 72 + \frac{- 21}{2} \cdot 8 + 28$

Step 2.4.3.1.3.2

Factor $2$ out of $8$ .

$y = 72 + \frac{- 21}{2} \cdot (2 (4)) + 28$

Step 2.4.3.1.3.3

Cancel the common factor.

$y = 72 + \frac{- 21}{2} \cdot (2 \cdot 4) + 28$

Step 2.4.3.1.3.4

Rewrite the expression.

$y = 72 - 21 \cdot 4 + 28$

Step 2.4.3.1.4

Multiply $- 21$ by $4$ .

$y = 72 - 84 + 28$

Step 2.4.3.2

Simplify by adding and subtracting.

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Step 2.4.3.2.1

Subtract $84$ from $72$ .

$y = - 12 + 28$

Step 2.4.3.2.2

Add $- 12$ and $28$ .

$y = 16$

Step 2.4.4

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 16$ and $y = 16$ .

$16 = 16$

Step 2.4.5

Calculate the value of $y$ such that $y = a x^{2} + b$ when $a = \frac{9}{8}$ , $b = - \frac{21}{2}$ , $c = 28$ , and $x = 12$ .

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Step 2.4.5.1

Simplify each term.

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Step 2.4.5.1.1

Raise $12$ to the power of $2$ .

$y = \frac{9}{8} \cdot 144 + (- \frac{21}{2}) \cdot (12) + 28$

Step 2.4.5.1.2

Cancel the common factor of $8$ .

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Step 2.4.5.1.2.1

Factor $8$ out of $144$ .

$y = \frac{9}{8} \cdot (8 (18)) + (- \frac{21}{2}) \cdot (12) + 28$

Step 2.4.5.1.2.2

Cancel the common factor.

$y = \frac{9}{8} \cdot (8 \cdot 18) + (- \frac{21}{2}) \cdot (12) + 28$

Step 2.4.5.1.2.3

Rewrite the expression.

$y = 9 \cdot 18 + (- \frac{21}{2}) \cdot (12) + 28$

Step 2.4.5.1.3

Multiply $9$ by $18$ .

$y = 162 + (- \frac{21}{2}) \cdot (12) + 28$

Step 2.4.5.1.4

Cancel the common factor of $2$ .

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Step 2.4.5.1.4.1

Move the leading negative in $- \frac{21}{2}$ into the numerator.

$y = 162 + \frac{- 21}{2} \cdot 12 + 28$

Step 2.4.5.1.4.2

Factor $2$ out of $12$ .

$y = 162 + \frac{- 21}{2} \cdot (2 (6)) + 28$

Step 2.4.5.1.4.3

Cancel the common factor.

$y = 162 + \frac{- 21}{2} \cdot (2 \cdot 6) + 28$

Step 2.4.5.1.4.4

Rewrite the expression.

$y = 162 - 21 \cdot 6 + 28$

Step 2.4.5.1.5

Multiply $- 21$ by $6$ .

$y = 162 - 126 + 28$

Step 2.4.5.2

Simplify by adding and subtracting.

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Step 2.4.5.2.1

Subtract $126$ from $162$ .

$y = 36 + 28$

Step 2.4.5.2.2

Add $36$ and $28$ .

$y = 64$

Step 2.4.6

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 12$ . This check passes since $y = 64$ and $y = 64$ .

$64 = 64$

Step 2.4.7

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

The function is quadratic

Step 3

Since all $y = y$ , the function is quadratic and follows the form $y = \frac{9 x^{2}}{8} - \frac{21 x}{2} + 28$ .

$y = \frac{9 x^{2}}{8} - \frac{21 x}{2} + 28$