Algebra Examples

$\begin{array}{c} x & y \\ 2 & - 18 \\ 6 & - 27 \\ 10 & - 36 \end{array}$

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 $y = a x + b$ .

$- 18 = a (2) + b - 27 = a (6) + b - 36 = a (10) + b$

Step 1.3

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

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Step 1.3.1

Solve for $b$ in $- 18 = a (2) + b$ .

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Step 1.3.1.1

Rewrite the equation as $a (2) + b = - 18$ .

$a (2) + b = - 18$

$- 27 = a (6) + b$

$- 36 = a (10) + b$

Step 1.3.1.2

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

$2 a + b = - 18$

$- 27 = a (6) + b$

$- 36 = a (10) + b$

Step 1.3.1.3

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

$b = - 18 - 2 a$

$- 27 = a (6) + b$

$- 36 = a (10) + b$

$b = - 18 - 2 a$

$- 27 = a (6) + b$

$- 36 = a (10) + b$

Step 1.3.2

Replace all occurrences of $b$ with $- 18 - 2 a$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $- 27 = a (6) + b$ with $- 18 - 2 a$ .

$- 27 = a (6) - 18 - 2 a$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

Step 1.3.2.2

Simplify $- 27 = a (6) - 18 - 2 a$ .

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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.

$- 27 = a (6) - 18 - 2 a$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

$- 27 = a (6) - 18 - 2 a$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

Simplify $a (6) - 18 - 2 a$ .

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Step 1.3.2.2.2.1.1

Move $6$ to the left of $a$ .

$- 27 = 6 a - 18 - 2 a$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

Step 1.3.2.2.2.1.2

Subtract $2 a$ from $6 a$ .

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = a (10) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $- 36 = a (10) + b$ with $- 18 - 2 a$ .

$- 36 = a (10) - 18 - 2 a$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.2.4

Simplify $- 36 = a (10) - 18 - 2 a$ .

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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.

$- 36 = a (10) - 18 - 2 a$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = a (10) - 18 - 2 a$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

Simplify $a (10) - 18 - 2 a$ .

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Step 1.3.2.4.2.1.1

Move $10$ to the left of $a$ .

$- 36 = 10 a - 18 - 2 a$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.2.4.2.1.2

Subtract $2 a$ from $10 a$ .

$- 36 = 8 a - 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = 8 a - 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = 8 a - 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = 8 a - 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$- 36 = 8 a - 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.3

Solve for $a$ in $- 36 = 8 a - 18$ .

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Step 1.3.3.1

Rewrite the equation as $8 a - 18 = - 36$ .

$8 a - 18 = - 36$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

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

Add $18$ to both sides of the equation.

$8 a = - 36 + 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.3.2.2

Add $- 36$ and $18$ .

$8 a = - 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$8 a = - 18$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.3.3

Divide each term in $8 a = - 18$ by $8$ and simplify.

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Step 1.3.3.3.1

Divide each term in $8 a = - 18$ by $8$ .

$\frac{8 a}{8} = \frac{- 18}{8}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

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.

$\frac{8 a}{8} = \frac{- 18}{8}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 18}{8}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$a = \frac{- 18}{8}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$a = \frac{- 18}{8}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

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 $- 18$ and $8$ .

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Step 1.3.3.3.3.1.1

Factor $2$ out of $- 18$ .

$a = \frac{2 (- 9)}{8}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

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 $2$ out of $8$ .

$a = \frac{2 \cdot - 9}{2 \cdot 4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.3.3.3.1.2.2

Cancel the common factor.

$a = \frac{2 \cdot - 9}{2 \cdot 4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.3.3.3.1.2.3

Rewrite the expression.

$a = \frac{- 9}{4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$a = \frac{- 9}{4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$a = \frac{- 9}{4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.3.3.3.2

Move the negative in front of the fraction.

$a = - \frac{9}{4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$a = - \frac{9}{4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$a = - \frac{9}{4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

$a = - \frac{9}{4}$

$- 27 = 4 a - 18$

$b = - 18 - 2 a$

Step 1.3.4

Replace all occurrences of $a$ with $- \frac{9}{4}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $a$ in $- 27 = 4 a - 18$ with $- \frac{9}{4}$ .

$- 27 = 4 (- \frac{9}{4}) - 18$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $4 (- \frac{9}{4}) - 18$ .

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Step 1.3.4.2.1.1

Cancel the common factor of $4$ .

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Step 1.3.4.2.1.1.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$- 27 = 4 (\frac{- 9}{4}) - 18$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

Step 1.3.4.2.1.1.2

Cancel the common factor.

$- 27 = 4 (\frac{- 9}{4}) - 18$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

Step 1.3.4.2.1.1.3

Rewrite the expression.

$- 27 = - 9 - 18$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

$- 27 = - 9 - 18$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

Step 1.3.4.2.1.2

Subtract $18$ from $- 9$ .

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - 18 - 2 a$

Step 1.3.4.3

Replace all occurrences of $a$ in $b = - 18 - 2 a$ with $- \frac{9}{4}$ .

$b = - 18 - 2 (- \frac{9}{4})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $- 18 - 2 (- \frac{9}{4})$ .

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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

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$b = - 18 - 2 (\frac{- 9}{4})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.1.1.2

Factor $2$ out of $- 2$ .

$b = - 18 + 2 (- 1) (\frac{- 9}{4})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.1.1.3

Factor $2$ out of $4$ .

$b = - 18 + 2 \cdot (- 1 \frac{- 9}{2 \cdot 2})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.1.1.4

Cancel the common factor.

$b = - 18 + 2 \cdot (- 1 \frac{- 9}{2 \cdot 2})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.1.1.5

Rewrite the expression.

$b = - 18 - 1 (\frac{- 9}{2})$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - 18 - 1 (\frac{- 9}{2})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.1.2

Move the negative in front of the fraction.

$b = - 18 - 1 (- \frac{9}{2})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.1.3

Multiply $- 1 (- \frac{9}{2})$ .

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Step 1.3.4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$b = - 18 + 1 (\frac{9}{2})$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.1.3.2

Multiply $\frac{9}{2}$ by $1$ .

$b = - 18 + \frac{9}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - 18 + \frac{9}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - 18 + \frac{9}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.2

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = - 18 \cdot \frac{2}{2} + \frac{9}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.3

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

$b = \frac{- 18 \cdot 2}{2} + \frac{9}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.4

Combine the numerators over the common denominator.

$b = \frac{- 18 \cdot 2 + 9}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.5

Simplify the numerator.

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Step 1.3.4.4.1.5.1

Multiply $- 18$ by $2$ .

$b = \frac{- 36 + 9}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.5.2

Add $- 36$ and $9$ .

$b = \frac{- 27}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = \frac{- 27}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.4.4.1.6

Move the negative in front of the fraction.

$b = - \frac{27}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - \frac{27}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - \frac{27}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

$b = - \frac{27}{2}$

$- 27 = - 27$

$a = - \frac{9}{4}$

Step 1.3.5

Remove any equations from the system that are always true.

$b = - \frac{27}{2}$

$a = - \frac{9}{4}$

Step 1.3.6

List all of the solutions.

$b = - \frac{27}{2}, a = - \frac{9}{4}$

Step 1.4

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

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Step 1.4.1

Calculate the value of $y$ when $a = - \frac{9}{4}$ , $b = - \frac{27}{2}$ , and $x = 2$ .

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Step 1.4.1.1

Simplify each term.

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Step 1.4.1.1.1

Cancel the common factor of $2$ .

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Step 1.4.1.1.1.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$y = \frac{- 9}{4} \cdot 2 - \frac{27}{2}$

Step 1.4.1.1.1.2

Factor $2$ out of $4$ .

$y = \frac{- 9}{2 (2)} \cdot 2 - \frac{27}{2}$

Step 1.4.1.1.1.3

Cancel the common factor.

$y = \frac{- 9}{2 \cdot 2} \cdot 2 - \frac{27}{2}$

Step 1.4.1.1.1.4

Rewrite the expression.

$y = \frac{- 9}{2} - \frac{27}{2}$

Step 1.4.1.1.2

Move the negative in front of the fraction.

$y = - \frac{9}{2} - \frac{27}{2}$

Step 1.4.1.2

Combine fractions.

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Step 1.4.1.2.1

Combine the numerators over the common denominator.

$y = \frac{- 9 - 27}{2}$

Step 1.4.1.2.2

Simplify the expression.

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Step 1.4.1.2.2.1

Subtract $27$ from $- 9$ .

$y = \frac{- 36}{2}$

Step 1.4.1.2.2.2

Divide $- 36$ by $2$ .

$y = - 18$

Step 1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = - 18$ and $y = - 18$ .

$- 18 = - 18$

Step 1.4.3

Calculate the value of $y$ when $a = - \frac{9}{4}$ , $b = - \frac{27}{2}$ , and $x = 6$ .

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Cancel the common factor of $2$ .

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Step 1.4.3.1.1.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$y = \frac{- 9}{4} \cdot 6 - \frac{27}{2}$

Step 1.4.3.1.1.2

Factor $2$ out of $4$ .

$y = \frac{- 9}{2 (2)} \cdot 6 - \frac{27}{2}$

Step 1.4.3.1.1.3

Factor $2$ out of $6$ .

$y = \frac{- 9}{2 \cdot 2} \cdot (2 \cdot 3) - \frac{27}{2}$

Step 1.4.3.1.1.4

Cancel the common factor.

$y = \frac{- 9}{2 \cdot 2} \cdot (2 \cdot 3) - \frac{27}{2}$

Step 1.4.3.1.1.5

Rewrite the expression.

$y = \frac{- 9}{2} \cdot 3 - \frac{27}{2}$

Step 1.4.3.1.2

Combine $\frac{- 9}{2}$ and $3$ .

$y = \frac{- 9 \cdot 3}{2} - \frac{27}{2}$

Step 1.4.3.1.3

Multiply $- 9$ by $3$ .

$y = \frac{- 27}{2} - \frac{27}{2}$

Step 1.4.3.1.4

Move the negative in front of the fraction.

$y = - \frac{27}{2} - \frac{27}{2}$

Step 1.4.3.2

Combine fractions.

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Step 1.4.3.2.1

Combine the numerators over the common denominator.

$y = \frac{- 27 - 27}{2}$

Step 1.4.3.2.2

Simplify the expression.

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Step 1.4.3.2.2.1

Subtract $27$ from $- 27$ .

$y = \frac{- 54}{2}$

Step 1.4.3.2.2.2

Divide $- 54$ by $2$ .

$y = - 27$

Step 1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 6$ . This check passes since $y = - 27$ and $y = - 27$ .

$- 27 = - 27$

Step 1.4.5

Calculate the value of $y$ when $a = - \frac{9}{4}$ , $b = - \frac{27}{2}$ , and $x = 10$ .

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Step 1.4.5.1

Simplify each term.

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Step 1.4.5.1.1

Cancel the common factor of $2$ .

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Step 1.4.5.1.1.1

Move the leading negative in $- \frac{9}{4}$ into the numerator.

$y = \frac{- 9}{4} \cdot 10 - \frac{27}{2}$

Step 1.4.5.1.1.2

Factor $2$ out of $4$ .

$y = \frac{- 9}{2 (2)} \cdot 10 - \frac{27}{2}$

Step 1.4.5.1.1.3

Factor $2$ out of $10$ .

$y = \frac{- 9}{2 \cdot 2} \cdot (2 \cdot 5) - \frac{27}{2}$

Step 1.4.5.1.1.4

Cancel the common factor.

$y = \frac{- 9}{2 \cdot 2} \cdot (2 \cdot 5) - \frac{27}{2}$

Step 1.4.5.1.1.5

Rewrite the expression.

$y = \frac{- 9}{2} \cdot 5 - \frac{27}{2}$

Step 1.4.5.1.2

Combine $\frac{- 9}{2}$ and $5$ .

$y = \frac{- 9 \cdot 5}{2} - \frac{27}{2}$

Step 1.4.5.1.3

Multiply $- 9$ by $5$ .

$y = \frac{- 45}{2} - \frac{27}{2}$

Step 1.4.5.1.4

Move the negative in front of the fraction.

$y = - \frac{45}{2} - \frac{27}{2}$

Step 1.4.5.2

Combine fractions.

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Step 1.4.5.2.1

Combine the numerators over the common denominator.

$y = \frac{- 45 - 27}{2}$

Step 1.4.5.2.2

Simplify the expression.

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Step 1.4.5.2.2.1

Subtract $27$ from $- 45$ .

$y = \frac{- 72}{2}$

Step 1.4.5.2.2.2

Divide $- 72$ by $2$ .

$y = - 36$

Step 1.4.6

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 10$ . This check passes since $y = - 36$ and $y = - 36$ .

$- 36 = - 36$

Step 1.4.7

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

The function is linear

Step 2

Since all $y = y$ , the function is linear and follows the form $y = - \frac{9 x}{4} - \frac{27}{2}$ .

$y = - \frac{9 x}{4} - \frac{27}{2}$