Algebra Examples

$\begin{array}{c} x & y \\ 1 & 3.5 \\ 3 & - 3.5 \\ 4 & - 10 \end{array}$

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 $y = a x + b$ .

$3.5 = a (1) + b - 3.5 = a (3) + b - 10 = a (4) + b$

Step 1.3

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

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Step 1.3.1

Solve for $a$ in $3.5 = a + b$ .

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Step 1.3.1.1

Rewrite the equation as $a + b = 3.5$ .

$a + b = 3.5$

$- 3.5 = a (3) + b$

$- 10 = a (4) + b$

Step 1.3.1.2

Subtract $b$ from both sides of the equation.

$a = 3.5 - b$

$- 3.5 = a (3) + b$

$- 10 = a (4) + b$

$a = 3.5 - b$

$- 3.5 = a (3) + b$

$- 10 = a (4) + b$

Step 1.3.2

Replace all occurrences of $a$ with $3.5 - b$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $a$ in $- 3.5 = a (3) + b$ with $3.5 - b$ .

$- 3.5 = (3.5 - b) (3) + b$

$a = 3.5 - b$

$- 10 = a (4) + b$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify $(3.5 - b) (3) + b$ .

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Step 1.3.2.2.1.1

Simplify each term.

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Step 1.3.2.2.1.1.1

Apply the distributive property.

$- 3.5 = 3.5 \cdot 3 - b \cdot 3 + b$

$a = 3.5 - b$

$- 10 = a (4) + b$

Step 1.3.2.2.1.1.2

Multiply $3.5$ by $3$ .

$- 3.5 = 10.5 - b \cdot 3 + b$

$a = 3.5 - b$

$- 10 = a (4) + b$

Step 1.3.2.2.1.1.3

Multiply $3$ by $- 1$ .

$- 3.5 = 10.5 - 3 b + b$

$a = 3.5 - b$

$- 10 = a (4) + b$

$- 3.5 = 10.5 - 3 b + b$

$a = 3.5 - b$

$- 10 = a (4) + b$

Step 1.3.2.2.1.2

Add $- 3 b$ and $b$ .

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 10 = a (4) + b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 10 = a (4) + b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 10 = a (4) + b$

Step 1.3.2.3

Replace all occurrences of $a$ in $- 10 = a (4) + b$ with $3.5 - b$ .

$- 10 = (3.5 - b) (4) + b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.2.4

Simplify the right side.

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Step 1.3.2.4.1

Simplify $(3.5 - b) (4) + b$ .

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Step 1.3.2.4.1.1

Simplify each term.

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Step 1.3.2.4.1.1.1

Apply the distributive property.

$- 10 = 3.5 \cdot 4 - b \cdot 4 + b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.2.4.1.1.2

Multiply $3.5$ by $4$ .

$- 10 = 14 - b \cdot 4 + b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.2.4.1.1.3

Multiply $4$ by $- 1$ .

$- 10 = 14 - 4 b + b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 10 = 14 - 4 b + b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.2.4.1.2

Add $- 4 b$ and $b$ .

$- 10 = 14 - 3 b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 10 = 14 - 3 b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 10 = 14 - 3 b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 10 = 14 - 3 b$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.3

Solve for $b$ in $- 10 = 14 - 3 b$ .

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Step 1.3.3.1

Rewrite the equation as $14 - 3 b = - 10$ .

$14 - 3 b = - 10$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Subtract $14$ from both sides of the equation.

$- 3 b = - 10 - 14$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.3.2.2

Subtract $14$ from $- 10$ .

$- 3 b = - 24$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$- 3 b = - 24$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.3.3

Divide each term in $- 3 b = - 24$ by $- 3$ and simplify.

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Step 1.3.3.3.1

Divide each term in $- 3 b = - 24$ by $- 3$ .

$\frac{- 3 b}{- 3} = \frac{- 24}{- 3}$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

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 $- 3$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 3 b}{- 3} = \frac{- 24}{- 3}$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 24}{- 3}$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$b = \frac{- 24}{- 3}$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$b = \frac{- 24}{- 3}$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $- 24$ by $- 3$ .

$b = 8$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$b = 8$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$b = 8$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

$b = 8$

$- 3.5 = 10.5 - 2 b$

$a = 3.5 - b$

Step 1.3.4

Replace all occurrences of $b$ with $8$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $b$ in $- 3.5 = 10.5 - 2 b$ with $8$ .

$- 3.5 = 10.5 - 2 \cdot 8$

$b = 8$

$a = 3.5 - b$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $10.5 - 2 \cdot 8$ .

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Step 1.3.4.2.1.1

Multiply $- 2$ by $8$ .

$- 3.5 = 10.5 - 16$

$b = 8$

$a = 3.5 - b$

Step 1.3.4.2.1.2

Subtract $16$ from $10.5$ .

$- 3.5 = - 5.5$

$b = 8$

$a = 3.5 - b$

$- 3.5 = - 5.5$

$b = 8$

$a = 3.5 - b$

$- 3.5 = - 5.5$

$b = 8$

$a = 3.5 - b$

Step 1.3.4.3

Replace all occurrences of $b$ in $a = 3.5 - b$ with $8$ .

$a = 3.5 - (8)$

$- 3.5 = - 5.5$

$b = 8$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $3.5 - (8)$ .

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Step 1.3.4.4.1.1

Multiply $- 1$ by $8$ .

$a = 3.5 - 8$

$- 3.5 = - 5.5$

$b = 8$

Step 1.3.4.4.1.2

Subtract $8$ from $3.5$ .

$a = - 4.5$

$- 3.5 = - 5.5$

$b = 8$

$a = - 4.5$

$- 3.5 = - 5.5$

$b = 8$

$a = - 4.5$

$- 3.5 = - 5.5$

$b = 8$

$a = - 4.5$

$- 3.5 = - 5.5$

$b = 8$

Step 1.3.5

Since $- 3.5 = - 5.5$ 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 $a$ in $3.5 = a + b + c$ .

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Step 2.3.1.1

Rewrite the equation as $a + b + c = 3.5$ .

$a + b + c = 3.5$

$- 3.5 = a \cdot 3^{2} + b (3) + c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Subtract $b$ from both sides of the equation.

$a + c = 3.5 - b$

$- 3.5 = a \cdot 3^{2} + b (3) + c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.1.2.2

Subtract $c$ from both sides of the equation.

$a = 3.5 - b - c$

$- 3.5 = a \cdot 3^{2} + b (3) + c$

$- 10 = a \cdot 4^{2} + b (4) + c$

$a = 3.5 - b - c$

$- 3.5 = a \cdot 3^{2} + b (3) + c$

$- 10 = a \cdot 4^{2} + b (4) + c$

$a = 3.5 - b - c$

$- 3.5 = a \cdot 3^{2} + b (3) + c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2

Replace all occurrences of $a$ with $3.5 - b - c$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $a$ in $- 3.5 = a \cdot 3^{2} + b (3) + c$ with $3.5 - b - c$ .

$- 3.5 = (3.5 - b - c) \cdot 3^{2} + b (3) + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $(3.5 - b - c) \cdot 3^{2} + b (3) + c$ .

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Step 2.3.2.2.1.1

Simplify each term.

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Step 2.3.2.2.1.1.1

Raise $3$ to the power of $2$ .

$- 3.5 = (3.5 - b - c) \cdot 9 + b (3) + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2.1.1.2

Apply the distributive property.

$- 3.5 = 3.5 \cdot 9 - b \cdot 9 - c \cdot 9 + b (3) + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2.1.1.3

Simplify.

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Step 2.3.2.2.1.1.3.1

Multiply $3.5$ by $9$ .

$- 3.5 = 31.5 - b \cdot 9 - c \cdot 9 + b (3) + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2.1.1.3.2

Multiply $9$ by $- 1$ .

$- 3.5 = 31.5 - 9 b - c \cdot 9 + b (3) + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2.1.1.3.3

Multiply $9$ by $- 1$ .

$- 3.5 = 31.5 - 9 b - 9 c + b (3) + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

$- 3.5 = 31.5 - 9 b - 9 c + b (3) + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2.1.1.4

Move $3$ to the left of $b$ .

$- 3.5 = 31.5 - 9 b - 9 c + 3 b + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

$- 3.5 = 31.5 - 9 b - 9 c + 3 b + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2.1.2

Simplify by adding terms.

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Step 2.3.2.2.1.2.1

Add $- 9 b$ and $3 b$ .

$- 3.5 = 31.5 - 6 b - 9 c + c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.2.1.2.2

Add $- 9 c$ and $c$ .

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = a \cdot 4^{2} + b (4) + c$

Step 2.3.2.3

Replace all occurrences of $a$ in $- 10 = a \cdot 4^{2} + b (4) + c$ with $3.5 - b - c$ .

$- 10 = (3.5 - b - c) \cdot 4^{2} + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4

Simplify the right side.

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Step 2.3.2.4.1

Simplify $(3.5 - b - c) \cdot 4^{2} + b (4) + c$ .

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Step 2.3.2.4.1.1

Simplify each term.

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Step 2.3.2.4.1.1.1

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

$- 10 = (3.5 - b - c) \cdot 16 + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4.1.1.2

Apply the distributive property.

$- 10 = 3.5 \cdot 16 - b \cdot 16 - c \cdot 16 + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4.1.1.3

Simplify.

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Step 2.3.2.4.1.1.3.1

Multiply $3.5$ by $16$ .

$- 10 = 56 - b \cdot 16 - c \cdot 16 + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4.1.1.3.2

Multiply $16$ by $- 1$ .

$- 10 = 56 - 16 b - c \cdot 16 + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4.1.1.3.3

Multiply $16$ by $- 1$ .

$- 10 = 56 - 16 b - 16 c + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = 56 - 16 b - 16 c + b (4) + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4.1.1.4

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

$- 10 = 56 - 16 b - 16 c + 4 b + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = 56 - 16 b - 16 c + 4 b + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4.1.2

Simplify by adding terms.

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Step 2.3.2.4.1.2.1

Add $- 16 b$ and $4 b$ .

$- 10 = 56 - 12 b - 16 c + c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.2.4.1.2.2

Add $- 16 c$ and $c$ .

$- 10 = 56 - 12 b - 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = 56 - 12 b - 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = 56 - 12 b - 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = 56 - 12 b - 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 10 = 56 - 12 b - 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3

Solve for $b$ in $- 10 = 56 - 12 b - 15 c$ .

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Step 2.3.3.1

Rewrite the equation as $56 - 12 b - 15 c = - 10$ .

$56 - 12 b - 15 c = - 10$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.2

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

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Step 2.3.3.2.1

Subtract $56$ from both sides of the equation.

$- 12 b - 15 c = - 10 - 56$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.2.2

Add $15 c$ to both sides of the equation.

$- 12 b = - 10 - 56 + 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.2.3

Subtract $56$ from $- 10$ .

$- 12 b = - 66 + 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$- 12 b = - 66 + 15 c$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3

Divide each term in $- 12 b = - 66 + 15 c$ by $- 12$ and simplify.

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Step 2.3.3.3.1

Divide each term in $- 12 b = - 66 + 15 c$ by $- 12$ .

$\frac{- 12 b}{- 12} = \frac{- 66}{- 12} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

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 $- 12$ .

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Step 2.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 12 b}{- 12} = \frac{- 66}{- 12} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 66}{- 12} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{- 66}{- 12} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{- 66}{- 12} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

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 $- 66$ and $- 12$ .

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Step 2.3.3.3.3.1.1.1

Factor $- 6$ out of $- 66$ .

$b = \frac{- 6 \cdot 11}{- 12} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

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 $- 6$ out of $- 12$ .

$b = \frac{- 6 \cdot 11}{- 6 \cdot 2} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.3.1.1.2.2

Cancel the common factor.

$b = \frac{- 6 \cdot 11}{- 6 \cdot 2} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.3.1.1.2.3

Rewrite the expression.

$b = \frac{11}{2} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} + \frac{15 c}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.3.1.2

Cancel the common factor of $15$ and $- 12$ .

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Step 2.3.3.3.3.1.2.1

Factor $3$ out of $15 c$ .

$b = \frac{11}{2} + \frac{3 (5 c)}{- 12}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.3.1.2.2

Cancel the common factors.

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Step 2.3.3.3.3.1.2.2.1

Factor $3$ out of $- 12$ .

$b = \frac{11}{2} + \frac{3 (5 c)}{3 (- 4)}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.3.1.2.2.2

Cancel the common factor.

$b = \frac{11}{2} + \frac{3 (5 c)}{3 \cdot - 4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.3.1.2.2.3

Rewrite the expression.

$b = \frac{11}{2} + \frac{5 c}{- 4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} + \frac{5 c}{- 4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} + \frac{5 c}{- 4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.3.3.3.1.3

Move the negative in front of the fraction.

$b = \frac{11}{2} - \frac{5 c}{4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$- 3.5 = 31.5 - 6 b - 8 c$

$a = 3.5 - b - c$

Step 2.3.4

Replace all occurrences of $b$ with $\frac{11}{2} - \frac{5 c}{4}$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $b$ in $- 3.5 = 31.5 - 6 b - 8 c$ with $\frac{11}{2} - \frac{5 c}{4}$ .

$- 3.5 = 31.5 - 6 (\frac{11}{2} - \frac{5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $31.5 - 6 (\frac{11}{2} - \frac{5 c}{4}) - 8 c$ .

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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.

$- 3.5 = 31.5 - 6 (\frac{11}{2}) - 6 (- \frac{5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.4.2.1.1.2.1

Factor $2$ out of $- 6$ .

$- 3.5 = 31.5 + 2 (- 3) (\frac{11}{2}) - 6 (- \frac{5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.2.2

Cancel the common factor.

$- 3.5 = 31.5 + 2 \cdot (- 3 (\frac{11}{2})) - 6 (- \frac{5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.2.3

Rewrite the expression.

$- 3.5 = 31.5 - 3 \cdot 11 - 6 (- \frac{5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = 31.5 - 3 \cdot 11 - 6 (- \frac{5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.3

Multiply $- 3$ by $11$ .

$- 3.5 = 31.5 - 33 - 6 (- \frac{5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.3.4.2.1.1.4.1

Move the leading negative in $- \frac{5 c}{4}$ into the numerator.

$- 3.5 = 31.5 - 33 - 6 \frac{- 5 c}{4} - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.4.2

Factor $2$ out of $- 6$ .

$- 3.5 = 31.5 - 33 + 2 (- 3) (\frac{- 5 c}{4}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.4.3

Factor $2$ out of $4$ .

$- 3.5 = 31.5 - 33 + 2 \cdot (- 3 \frac{- 5 c}{2 \cdot 2}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.4.4

Cancel the common factor.

$- 3.5 = 31.5 - 33 + 2 \cdot (- 3 \frac{- 5 c}{2 \cdot 2}) - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.4.5

Rewrite the expression.

$- 3.5 = 31.5 - 33 - 3 \frac{- 5 c}{2} - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = 31.5 - 33 - 3 \frac{- 5 c}{2} - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.5

Combine $- 3$ and $\frac{- 5 c}{2}$ .

$- 3.5 = 31.5 - 33 + \frac{- 3 (- 5 c)}{2} - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.1.6

Multiply $- 5$ by $- 3$ .

$- 3.5 = 31.5 - 33 + \frac{15 c}{2} - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = 31.5 - 33 + \frac{15 c}{2} - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.2

Subtract $33$ from $31.5$ .

$- 3.5 = - 1.5 + \frac{15 c}{2} - 8 c$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.3

To write $- 8 c$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 3.5 = - 1.5 + \frac{15 c}{2} - 8 c \cdot \frac{2}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.4

Simplify terms.

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Step 2.3.4.2.1.4.1

Combine $- 8 c$ and $\frac{2}{2}$ .

$- 3.5 = - 1.5 + \frac{15 c}{2} + \frac{- 8 c \cdot 2}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.4.2

Combine the numerators over the common denominator.

$- 3.5 = - 1.5 + \frac{15 c - 8 c \cdot 2}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = - 1.5 + \frac{15 c - 8 c \cdot 2}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.5

Simplify each term.

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Step 2.3.4.2.1.5.1

Simplify the numerator.

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Step 2.3.4.2.1.5.1.1

Factor $c$ out of $15 c - 8 c \cdot 2$ .

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Step 2.3.4.2.1.5.1.1.1

Factor $c$ out of $15 c$ .

$- 3.5 = - 1.5 + \frac{c \cdot 15 - 8 c \cdot 2}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.5.1.1.2

Factor $c$ out of $- 8 c \cdot 2$ .

$- 3.5 = - 1.5 + \frac{c \cdot 15 + c (- 8 \cdot 2)}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.5.1.1.3

Factor $c$ out of $c \cdot 15 + c (- 8 \cdot 2)$ .

$- 3.5 = - 1.5 + \frac{c (15 - 8 \cdot 2)}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = - 1.5 + \frac{c (15 - 8 \cdot 2)}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.5.1.2

Multiply $- 8$ by $2$ .

$- 3.5 = - 1.5 + \frac{c (15 - 16)}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.5.1.3

Subtract $16$ from $15$ .

$- 3.5 = - 1.5 + \frac{c \cdot - 1}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = - 1.5 + \frac{c \cdot - 1}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.5.2

Move $- 1$ to the left of $c$ .

$- 3.5 = - 1.5 + \frac{- 1 \cdot c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.2.1.5.3

Move the negative in front of the fraction.

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - b - c$

Step 2.3.4.3

Replace all occurrences of $b$ in $a = 3.5 - b - c$ with $\frac{11}{2} - \frac{5 c}{4}$ .

$a = 3.5 - (\frac{11}{2} - \frac{5 c}{4}) - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $3.5 - (\frac{11}{2} - \frac{5 c}{4}) - c$ .

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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.

$a = 3.5 - \frac{11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.1.2

Multiply $- - \frac{5 c}{4}$ .

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Step 2.3.4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$a = 3.5 - \frac{11}{2} + 1 (\frac{5 c}{4}) - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.1.2.2

Multiply $\frac{5 c}{4}$ by $1$ .

$a = 3.5 - \frac{11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - \frac{11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = 3.5 - \frac{11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.2

To write $3.5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = 3.5 \cdot \frac{2}{2} - \frac{11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.3

Combine $3.5$ and $\frac{2}{2}$ .

$a = \frac{3.5 \cdot 2}{2} - \frac{11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.4

Combine the numerators over the common denominator.

$a = \frac{3.5 \cdot 2 - 11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.5

Simplify the numerator.

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Step 2.3.4.4.1.5.1

Multiply $3.5$ by $2$ .

$a = \frac{7 - 11}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.5.2

Subtract $11$ from $7$ .

$a = \frac{- 4}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = \frac{- 4}{2} + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.6

Divide $- 4$ by $2$ .

$a = - 2 + \frac{5 c}{4} - c$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.7

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

$a = - 2 + \frac{5 c}{4} - c \cdot \frac{4}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.8

Simplify terms.

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Step 2.3.4.4.1.8.1

Combine $- c$ and $\frac{4}{4}$ .

$a = - 2 + \frac{5 c}{4} + \frac{- c \cdot 4}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.8.2

Combine the numerators over the common denominator.

$a = - 2 + \frac{5 c - c \cdot 4}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 2 + \frac{5 c - c \cdot 4}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.9

Simplify each term.

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Step 2.3.4.4.1.9.1

Simplify the numerator.

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Step 2.3.4.4.1.9.1.1

Factor $c$ out of $5 c - c \cdot 4$ .

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Step 2.3.4.4.1.9.1.1.1

Factor $c$ out of $5 c$ .

$a = - 2 + \frac{c \cdot 5 - c \cdot 4}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.9.1.1.2

Factor $c$ out of $- c \cdot 4$ .

$a = - 2 + \frac{c \cdot 5 + c (- 1 \cdot 4)}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.9.1.1.3

Factor $c$ out of $c \cdot 5 + c (- 1 \cdot 4)$ .

$a = - 2 + \frac{c (5 - 1 \cdot 4)}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 2 + \frac{c (5 - 1 \cdot 4)}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.9.1.2

Multiply $- 1$ by $4$ .

$a = - 2 + \frac{c (5 - 4)}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.9.1.3

Subtract $4$ from $5$ .

$a = - 2 + \frac{c \cdot 1}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 2 + \frac{c \cdot 1}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.4.4.1.9.2

Multiply $c$ by $1$ .

$a = - 2 + \frac{c}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 2 + \frac{c}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 2 + \frac{c}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 2 + \frac{c}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 2 + \frac{c}{4}$

$- 3.5 = - 1.5 - \frac{c}{2}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5

Solve for $c$ in $- 3.5 = - 1.5 - \frac{c}{2}$ .

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Step 2.3.5.1

Rewrite the equation as $- 1.5 - \frac{c}{2} = - 3.5$ .

$- 1.5 - \frac{c}{2} = - 3.5$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Add $1.5$ to both sides of the equation.

$- \frac{c}{2} = - 3.5 + 1.5$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.2.2

Add $- 3.5$ and $1.5$ .

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

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

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

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.3

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{c}{2}) = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.4

Simplify both sides of the equation.

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Step 2.3.5.4.1

Simplify the left side.

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Step 2.3.5.4.1.1

Simplify $- 2 (- \frac{c}{2})$ .

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Step 2.3.5.4.1.1.1

Cancel the common factor of $2$ .

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Step 2.3.5.4.1.1.1.1

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

$- 2 \frac{- c}{2} = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.4.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- c}{2}) = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.4.1.1.1.3

Cancel the common factor.

$2 \cdot (- 1 \frac{- c}{2}) = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.4.1.1.1.4

Rewrite the expression.

$c = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.4.1.1.2

Multiply.

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Step 2.3.5.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 c = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.4.1.1.2.2

Multiply $c$ by $1$ .

$c = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 2$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.5.4.2

Simplify the right side.

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Step 2.3.5.4.2.1

Multiply $- 2$ by $- 2$ .

$c = 4$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$c = 4$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$c = 4$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

$c = 4$

$a = - 2 + \frac{c}{4}$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.6

Replace all occurrences of $c$ with $4$ in each equation.

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Step 2.3.6.1

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

$a = - 2 + \frac{4}{4}$

$c = 4$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $- 2 + \frac{4}{4}$ .

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Step 2.3.6.2.1.1

Divide $4$ by $4$ .

$a = - 2 + 1$

$c = 4$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.6.2.1.2

Add $- 2$ and $1$ .

$a = - 1$

$c = 4$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 1$

$c = 4$

$b = \frac{11}{2} - \frac{5 c}{4}$

$a = - 1$

$c = 4$

$b = \frac{11}{2} - \frac{5 c}{4}$

Step 2.3.6.3

Replace all occurrences of $c$ in $b = \frac{11}{2} - \frac{5 c}{4}$ with $4$ .

$b = \frac{11}{2} - \frac{5 (4)}{4}$

$a = - 1$

$c = 4$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $\frac{11}{2} - \frac{5 (4)}{4}$ .

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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

Cancel the common factor of $4$ and $4$ .

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Step 2.3.6.4.1.1.1.1

Cancel the common factor.

$b = \frac{11}{2} - \frac{5 \cdot 4}{4}$

$a = - 1$

$c = 4$

Step 2.3.6.4.1.1.1.2

Divide $5$ by $1$ .

$b = \frac{11}{2} - 1 \cdot 5$

$a = - 1$

$c = 4$

$b = \frac{11}{2} - 1 \cdot 5$

$a = - 1$

$c = 4$

Step 2.3.6.4.1.1.2

Multiply $- 1$ by $5$ .

$b = \frac{11}{2} - 5$

$a = - 1$

$c = 4$

$b = \frac{11}{2} - 5$

$a = - 1$

$c = 4$

Step 2.3.6.4.1.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = \frac{11}{2} - 5 \cdot \frac{2}{2}$

$a = - 1$

$c = 4$

Step 2.3.6.4.1.3

Combine $- 5$ and $\frac{2}{2}$ .

$b = \frac{11}{2} + \frac{- 5 \cdot 2}{2}$

$a = - 1$

$c = 4$

Step 2.3.6.4.1.4

Combine the numerators over the common denominator.

$b = \frac{11 - 5 \cdot 2}{2}$

$a = - 1$

$c = 4$

Step 2.3.6.4.1.5

Simplify the numerator.

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Step 2.3.6.4.1.5.1

Multiply $- 5$ by $2$ .

$b = \frac{11 - 10}{2}$

$a = - 1$

$c = 4$

Step 2.3.6.4.1.5.2

Subtract $10$ from $11$ .

$b = \frac{1}{2}$

$a = - 1$

$c = 4$

$b = \frac{1}{2}$

$a = - 1$

$c = 4$

$b = \frac{1}{2}$

$a = - 1$

$c = 4$

$b = \frac{1}{2}$

$a = - 1$

$c = 4$

$b = \frac{1}{2}$

$a = - 1$

$c = 4$

Step 2.3.7

List all of the solutions.

$b = \frac{1}{2}, a = - 1, c = 4$

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 $a = - 1$ , $b = \frac{1}{2}$ , $c = 4$ , and $x = 1$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

One to any power is one.

$y = - 1 \cdot 1 + (\frac{1}{2}) \cdot (1) + 4$

Step 2.4.1.1.2

Multiply $- 1$ by $1$ .

$y = - 1 + (\frac{1}{2}) \cdot (1) + 4$

Step 2.4.1.1.3

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

$y = - 1 + \frac{1}{2} + 4$

Step 2.4.1.2

Find the common denominator.

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Step 2.4.1.2.1

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

$y = \frac{- 1}{1} + \frac{1}{2} + 4$

Step 2.4.1.2.2

Multiply $\frac{- 1}{1}$ by $\frac{2}{2}$ .

$y = \frac{- 1}{1} \cdot \frac{2}{2} + \frac{1}{2} + 4$

Step 2.4.1.2.3

Multiply $\frac{- 1}{1}$ by $\frac{2}{2}$ .

$y = \frac{- 1 \cdot 2}{2} + \frac{1}{2} + 4$

Step 2.4.1.2.4

Write $4$ as a fraction with denominator $1$ .

$y = \frac{- 1 \cdot 2}{2} + \frac{1}{2} + \frac{4}{1}$

Step 2.4.1.2.5

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

$y = \frac{- 1 \cdot 2}{2} + \frac{1}{2} + \frac{4}{1} \cdot \frac{2}{2}$

Step 2.4.1.2.6

Multiply $\frac{4}{1}$ by $\frac{2}{2}$ .

$y = \frac{- 1 \cdot 2}{2} + \frac{1}{2} + \frac{4 \cdot 2}{2}$

Step 2.4.1.3

Combine the numerators over the common denominator.

$y = \frac{- 1 \cdot 2 + 1 + 4 \cdot 2}{2}$

Step 2.4.1.4

Simplify each term.

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Step 2.4.1.4.1

Multiply $- 1$ by $2$ .

$y = \frac{- 2 + 1 + 4 \cdot 2}{2}$

Step 2.4.1.4.2

Multiply $4$ by $2$ .

$y = \frac{- 2 + 1 + 8}{2}$

Step 2.4.1.5

Reduce the expression by cancelling the common factors.

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Step 2.4.1.5.1

Add $- 2$ and $1$ .

$y = \frac{- 1 + 8}{2}$

Step 2.4.1.5.2

Add $- 1$ and $8$ .

$y = \frac{7}{2}$

Step 2.4.1.5.3

Cancel the common factor of $7$ and $2$ .

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Step 2.4.1.5.3.1

Rewrite $7$ as $1 (7)$ .

$y = \frac{1 (7)}{2}$

Step 2.4.1.5.3.2

Cancel the common factors.

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Step 2.4.1.5.3.2.1

Rewrite $2$ as $1 (2)$ .

$y = \frac{1 \cdot 7}{1 \cdot 2}$

Step 2.4.1.5.3.2.2

Cancel the common factor.

$y = \frac{1 \cdot 7}{1 \cdot 2}$

Step 2.4.1.5.3.2.3

Rewrite the expression.

$y = \frac{7}{2}$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check does not pass, since $y = \frac{7}{2}$ and $y = 3.5$ . The function rule can't be quadratic.

$\frac{7}{2} \neq 3.5$

Step 2.4.3

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

The function is not quadratic

Step 3

There are no values of $a$ , $b$ , or $c$ in the equations $y = a x + b$ or $y = a x^{2} + b x + c$ that work for every pair of $x$ and $y$ .

The table does not have a function rule that is linear or quadratic.