Algebra Examples

$\begin{array}{c} x & y \\ - 3 & 221.86 \\ - 2 & 37.71 \\ - 1 & 6.41 \end{array}$

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 $y = a x + b$ .

$221.86 = a (- 3) + b 37.71 = a (- 2) + b 6.41 = a (- 1) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $221.86 = a (- 3) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (- 3) + b = 221.86$ .

$a (- 3) + b = 221.86$

$37.71 = a (- 2) + b$

$6.41 = a (- 1) + b$

Step 1.3.1.2

Move $- 3$ to the left of $a$ .

$- 3 a + b = 221.86$

$37.71 = a (- 2) + b$

$6.41 = a (- 1) + b$

Step 1.3.1.3

Add $3 a$ to both sides of the equation.

$b = 221.86 + 3 a$

$37.71 = a (- 2) + b$

$6.41 = a (- 1) + b$

$b = 221.86 + 3 a$

$37.71 = a (- 2) + b$

$6.41 = a (- 1) + b$

Step 1.3.2

Replace all occurrences of $b$ with $221.86 + 3 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $37.71 = a (- 2) + b$ with $221.86 + 3 a$ .

$37.71 = a (- 2) + 221.86 + 3 a$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

Step 1.3.2.2

Simplify $37.71 = a (- 2) + 221.86 + 3 a$ .

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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.

$37.71 = a (- 2) + 221.86 + 3 a$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

$37.71 = a (- 2) + 221.86 + 3 a$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (- 2) + 221.86 + 3 a$ .

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Step 1.3.2.2.2.1.1

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

$37.71 = - 2 a + 221.86 + 3 a$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

Step 1.3.2.2.2.1.2

Add $- 2 a$ and $3 a$ .

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $6.41 = a (- 1) + b$ with $221.86 + 3 a$ .

$6.41 = a (- 1) + 221.86 + 3 a$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.2.4

Simplify $6.41 = a (- 1) + 221.86 + 3 a$ .

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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.

$6.41 = a (- 1) + 221.86 + 3 a$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = a (- 1) + 221.86 + 3 a$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (- 1) + 221.86 + 3 a$ .

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Step 1.3.2.4.2.1.1

Simplify each term.

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Step 1.3.2.4.2.1.1.1

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

$6.41 = - 1 a + 221.86 + 3 a$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.2.4.2.1.1.2

Rewrite $- 1 a$ as $- a$ .

$6.41 = - a + 221.86 + 3 a$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = - a + 221.86 + 3 a$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.2.4.2.1.2

Add $- a$ and $3 a$ .

$6.41 = 2 a + 221.86$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = 2 a + 221.86$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = 2 a + 221.86$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = 2 a + 221.86$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$6.41 = 2 a + 221.86$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.3

Solve for $a$ in $6.41 = 2 a + 221.86$ .

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Step 1.3.3.1

Rewrite the equation as $2 a + 221.86 = 6.41$ .

$2 a + 221.86 = 6.41$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

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 $221.86$ from both sides of the equation.

$2 a = 6.41 - 221.86$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.3.2.2

Subtract $221.86$ from $6.41$ .

$2 a = - 215.45$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$2 a = - 215.45$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.3.3

Divide each term in $2 a = - 215.45$ by $2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $2 a = - 215.45$ by $2$ .

$\frac{2 a}{2} = \frac{- 215.45}{2}$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

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 $2$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 215.45}{2}$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 215.45}{2}$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$a = \frac{- 215.45}{2}$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$a = \frac{- 215.45}{2}$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $- 215.45$ by $2$ .

$a = - 107.725$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$a = - 107.725$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$a = - 107.725$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

$a = - 107.725$

$37.71 = a + 221.86$

$b = 221.86 + 3 a$

Step 1.3.4

Replace all occurrences of $a$ with $- 107.725$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $37.71 = a + 221.86$ with $- 107.725$ .

$37.71 = (- 107.725) + 221.86$

$a = - 107.725$

$b = 221.86 + 3 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Add $- 107.725$ and $221.86$ .

$37.71 = 114.135$

$a = - 107.725$

$b = 221.86 + 3 a$

$37.71 = 114.135$

$a = - 107.725$

$b = 221.86 + 3 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = 221.86 + 3 a$ with $- 107.725$ .

$b = 221.86 + 3 (- 107.725)$

$37.71 = 114.135$

$a = - 107.725$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $221.86 + 3 (- 107.725)$ .

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Step 1.3.4.4.1.1

Multiply $3$ by $- 107.725$ .

$b = 221.86 - 323.175$

$37.71 = 114.135$

$a = - 107.725$

Step 1.3.4.4.1.2

Subtract $323.175$ from $221.86$ .

$b = - 101.315$

$37.71 = 114.135$

$a = - 107.725$

$b = - 101.315$

$37.71 = 114.135$

$a = - 107.725$

$b = - 101.315$

$37.71 = 114.135$

$a = - 107.725$

$b = - 101.315$

$37.71 = 114.135$

$a = - 107.725$

Step 1.3.5

Since $37.71 = 114.135$ 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 $221.86 = a {(- 3)}^{2} + b (- 3) + c$ .

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Step 2.3.1.1

Rewrite the equation as $a {(- 3)}^{2} + b (- 3) + c = 221.86$ .

$a {(- 3)}^{2} + b (- 3) + c = 221.86$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.1.2

Simplify each term.

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Step 2.3.1.2.1

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

$a \cdot 9 + b (- 3) + c = 221.86$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.1.2.2

Move $9$ to the left of $a$ .

$9 \cdot a + b (- 3) + c = 221.86$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.1.2.3

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

$9 a - 3 b + c = 221.86$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$9 a - 3 b + c = 221.86$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

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 $9 a$ from both sides of the equation.

$- 3 b + c = 221.86 - 9 a$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.1.3.2

Add $3 b$ to both sides of the equation.

$c = 221.86 - 9 a + 3 b$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$c = 221.86 - 9 a + 3 b$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$c = 221.86 - 9 a + 3 b$

$37.71 = a {(- 2)}^{2} + b (- 2) + c$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.2

Replace all occurrences of $c$ with $221.86 - 9 a + 3 b$ in each equation.

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Step 2.3.2.1

Replace all occurrences of $c$ in $37.71 = a {(- 2)}^{2} + b (- 2) + c$ with $221.86 - 9 a + 3 b$ .

$37.71 = a {(- 2)}^{2} + b (- 2) + 221.86 - 9 a + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.2.2

Simplify $37.71 = a {(- 2)}^{2} + b (- 2) + 221.86 - 9 a + 3 b$ .

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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.

$37.71 = a {(- 2)}^{2} + b (- 2) + 221.86 - 9 a + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$37.71 = a {(- 2)}^{2} + b (- 2) + 221.86 - 9 a + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.2.2.2

Simplify the right side.

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Step 2.3.2.2.2.1

Simplify $a {(- 2)}^{2} + b (- 2) + 221.86 - 9 a + 3 b$ .

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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 $- 2$ to the power of $2$ .

$37.71 = a \cdot 4 + b (- 2) + 221.86 - 9 a + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.2.2.2.1.1.2

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

$37.71 = 4 \cdot a + b (- 2) + 221.86 - 9 a + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.2.2.2.1.1.3

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

$37.71 = 4 a - 2 b + 221.86 - 9 a + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$37.71 = 4 a - 2 b + 221.86 - 9 a + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

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 $9 a$ from $4 a$ .

$37.71 = - 5 a - 2 b + 221.86 + 3 b$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.2.2.2.1.2.2

Add $- 2 b$ and $3 b$ .

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + c$

Step 2.3.2.3

Replace all occurrences of $c$ in $6.41 = a {(- 1)}^{2} + b (- 1) + c$ with $221.86 - 9 a + 3 b$ .

$6.41 = a {(- 1)}^{2} + b (- 1) + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.2.4

Simplify $6.41 = a {(- 1)}^{2} + b (- 1) + 221.86 - 9 a + 3 b$ .

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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.

$6.41 = a {(- 1)}^{2} + b (- 1) + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = a {(- 1)}^{2} + b (- 1) + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.2.4.2

Simplify the right side.

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Step 2.3.2.4.2.1

Simplify $a {(- 1)}^{2} + b (- 1) + 221.86 - 9 a + 3 b$ .

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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 $- 1$ to the power of $2$ .

$6.41 = a \cdot 1 + b (- 1) + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.2.4.2.1.1.2

Multiply $a$ by $1$ .

$6.41 = a + b (- 1) + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.2.4.2.1.1.3

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

$6.41 = a - 1 b + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.2.4.2.1.1.4

Rewrite $- 1 b$ as $- b$ .

$6.41 = a - b + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = a - b + 221.86 - 9 a + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

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 $9 a$ from $a$ .

$6.41 = - 8 a - b + 221.86 + 3 b$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.2.4.2.1.2.2

Add $- b$ and $3 b$ .

$6.41 = - 8 a + 2 b + 221.86$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = - 8 a + 2 b + 221.86$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = - 8 a + 2 b + 221.86$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = - 8 a + 2 b + 221.86$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = - 8 a + 2 b + 221.86$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

$6.41 = - 8 a + 2 b + 221.86$

$37.71 = - 5 a + b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.3

Solve for $b$ in $37.71 = - 5 a + b + 221.86$ .

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Step 2.3.3.1

Rewrite the equation as $- 5 a + b + 221.86 = 37.71$ .

$- 5 a + b + 221.86 = 37.71$

$6.41 = - 8 a + 2 b + 221.86$

$c = 221.86 - 9 a + 3 b$

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

Add $5 a$ to both sides of the equation.

$b + 221.86 = 37.71 + 5 a$

$6.41 = - 8 a + 2 b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.3.2.2

Subtract $221.86$ from both sides of the equation.

$b = 37.71 + 5 a - 221.86$

$6.41 = - 8 a + 2 b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.3.2.3

Subtract $221.86$ from $37.71$ .

$b = 5 a - 184.15$

$6.41 = - 8 a + 2 b + 221.86$

$c = 221.86 - 9 a + 3 b$

$b = 5 a - 184.15$

$6.41 = - 8 a + 2 b + 221.86$

$c = 221.86 - 9 a + 3 b$

$b = 5 a - 184.15$

$6.41 = - 8 a + 2 b + 221.86$

$c = 221.86 - 9 a + 3 b$

Step 2.3.4

Replace all occurrences of $b$ with $5 a - 184.15$ in each equation.

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Step 2.3.4.1

Replace all occurrences of $b$ in $6.41 = - 8 a + 2 b + 221.86$ with $5 a - 184.15$ .

$6.41 = - 8 a + 2 (5 a - 184.15) + 221.86$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

Step 2.3.4.2

Simplify the right side.

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Step 2.3.4.2.1

Simplify $- 8 a + 2 (5 a - 184.15) + 221.86$ .

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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.

$6.41 = - 8 a + 2 (5 a) + 2 \cdot - 184.15 + 221.86$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

Step 2.3.4.2.1.1.2

Multiply $5$ by $2$ .

$6.41 = - 8 a + 10 a + 2 \cdot - 184.15 + 221.86$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

Step 2.3.4.2.1.1.3

Multiply $2$ by $- 184.15$ .

$6.41 = - 8 a + 10 a - 368.3 + 221.86$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

$6.41 = - 8 a + 10 a - 368.3 + 221.86$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

Step 2.3.4.2.1.2

Simplify by adding terms.

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Step 2.3.4.2.1.2.1

Add $- 8 a$ and $10 a$ .

$6.41 = 2 a - 368.3 + 221.86$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

Step 2.3.4.2.1.2.2

Add $- 368.3$ and $221.86$ .

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 3 b$

Step 2.3.4.3

Replace all occurrences of $b$ in $c = 221.86 - 9 a + 3 b$ with $5 a - 184.15$ .

$c = 221.86 - 9 a + 3 (5 a - 184.15)$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

Step 2.3.4.4

Simplify the right side.

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Step 2.3.4.4.1

Simplify $221.86 - 9 a + 3 (5 a - 184.15)$ .

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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.

$c = 221.86 - 9 a + 3 (5 a) + 3 \cdot - 184.15$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

Step 2.3.4.4.1.1.2

Multiply $5$ by $3$ .

$c = 221.86 - 9 a + 15 a + 3 \cdot - 184.15$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

Step 2.3.4.4.1.1.3

Multiply $3$ by $- 184.15$ .

$c = 221.86 - 9 a + 15 a - 552.45$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 221.86 - 9 a + 15 a - 552.45$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

Step 2.3.4.4.1.2

Simplify by adding terms.

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Step 2.3.4.4.1.2.1

Subtract $552.45$ from $221.86$ .

$c = - 9 a + 15 a - 330.59$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

Step 2.3.4.4.1.2.2

Add $- 9 a$ and $15 a$ .

$c = 6 a - 330.59$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 6 a - 330.59$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 6 a - 330.59$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 6 a - 330.59$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

$c = 6 a - 330.59$

$6.41 = 2 a - 146.44$

$b = 5 a - 184.15$

Step 2.3.5

Solve for $a$ in $6.41 = 2 a - 146.44$ .

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Step 2.3.5.1

Rewrite the equation as $2 a - 146.44 = 6.41$ .

$2 a - 146.44 = 6.41$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

Step 2.3.5.2

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

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Step 2.3.5.2.1

Add $146.44$ to both sides of the equation.

$2 a = 6.41 + 146.44$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

Step 2.3.5.2.2

Add $6.41$ and $146.44$ .

$2 a = 152.85$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

$2 a = 152.85$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

Step 2.3.5.3

Divide each term in $2 a = 152.85$ by $2$ and simplify.

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Step 2.3.5.3.1

Divide each term in $2 a = 152.85$ by $2$ .

$\frac{2 a}{2} = \frac{152.85}{2}$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

Step 2.3.5.3.2

Simplify the left side.

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Step 2.3.5.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.5.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{152.85}{2}$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

Step 2.3.5.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{152.85}{2}$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

$a = \frac{152.85}{2}$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

$a = \frac{152.85}{2}$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

Step 2.3.5.3.3

Simplify the right side.

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Step 2.3.5.3.3.1

Divide $152.85$ by $2$ .

$a = 76.425$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

$a = 76.425$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

$a = 76.425$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

$a = 76.425$

$c = 6 a - 330.59$

$b = 5 a - 184.15$

Step 2.3.6

Replace all occurrences of $a$ with $76.425$ in each equation.

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Step 2.3.6.1

Replace all occurrences of $a$ in $c = 6 a - 330.59$ with $76.425$ .

$c = 6 (76.425) - 330.59$

$a = 76.425$

$b = 5 a - 184.15$

Step 2.3.6.2

Simplify the right side.

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Step 2.3.6.2.1

Simplify $6 (76.425) - 330.59$ .

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Step 2.3.6.2.1.1

Multiply $6$ by $76.425$ .

$c = 458.55 - 330.59$

$a = 76.425$

$b = 5 a - 184.15$

Step 2.3.6.2.1.2

Subtract $330.59$ from $458.55$ .

$c = 127.96$

$a = 76.425$

$b = 5 a - 184.15$

$c = 127.96$

$a = 76.425$

$b = 5 a - 184.15$

$c = 127.96$

$a = 76.425$

$b = 5 a - 184.15$

Step 2.3.6.3

Replace all occurrences of $a$ in $b = 5 a - 184.15$ with $76.425$ .

$b = 5 (76.425) - 184.15$

$c = 127.96$

$a = 76.425$

Step 2.3.6.4

Simplify the right side.

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Step 2.3.6.4.1

Simplify $5 (76.425) - 184.15$ .

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Step 2.3.6.4.1.1

Multiply $5$ by $76.425$ .

$b = 382.125 - 184.15$

$c = 127.96$

$a = 76.425$

Step 2.3.6.4.1.2

Subtract $184.15$ from $382.125$ .

$b = 197.975$

$c = 127.96$

$a = 76.425$

$b = 197.975$

$c = 127.96$

$a = 76.425$

$b = 197.975$

$c = 127.96$

$a = 76.425$

$b = 197.975$

$c = 127.96$

$a = 76.425$

Step 2.3.7

List all of the solutions.

$b = 197.975, c = 127.96, a = 76.425$

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 $a = 76.425$ , $b = 197.975$ , $c = 127.96$ , and $x = - 3$ .

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Step 2.4.1.1

Simplify each term.

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Step 2.4.1.1.1

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

$y = 76.425 \cdot 9 + (197.975) \cdot (- 3) + 127.96$

Step 2.4.1.1.2

Multiply $76.425$ by $9$ .

$y = 687.825 + (197.975) \cdot (- 3) + 127.96$

Step 2.4.1.1.3

Multiply $197.975$ by $- 3$ .

$y = 687.825 - 593.925 + 127.96$

Step 2.4.1.2

Simplify by adding and subtracting.

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Step 2.4.1.2.1

Subtract $593.925$ from $687.825$ .

$y = 93.9 + 127.96$

Step 2.4.1.2.2

Add $93.9$ and $127.96$ .

$y = 221.86$

Step 2.4.2

If the table has a quadratic function rule, $y = y$ for the corresponding $x$ value, $x = - 3$ . This check does not pass, since $y = 221.86$ and $y = 221.86$ . The function rule can't be quadratic.

$221.86 \neq 221.86$

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.