Pre-Algebra Examples

$\begin{array}{c} x & y \\ - 3 & 5 \\ - 2 & 3 \\ - 1 & 1 \\ 0 & - 1 \\ 1 & - 3 \\ 2 & - 5 \end{array}$

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 $y = a x + b$ .

$5 = a (- 3) + b 3 = a (- 2) + b 1 = a (- 1) + b - 1 = a (0) + b - 3 = a (1) + b - 5 = a (2) + b$

Step 1.3

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

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Step 1.3.1

Rewrite the equation as $b = - 1$ .

$b = - 1$

$5 = a (- 3) + b$

$3 = a (- 2) + b$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2

Replace all occurrences of $b$ with $- 1$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $b$ in $5 = a (- 3) + b$ with $- 1$ .

$5 = a (- 3) - 1$

$b = - 1$

$3 = a (- 2) + b$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.2

Simplify $5 = a (- 3) - 1$ .

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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.

$5 = a (- 3) - 1$

$b = - 1$

$3 = a (- 2) + b$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

$5 = a (- 3) - 1$

$b = - 1$

$3 = a (- 2) + b$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.2.2

Simplify the right side.

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Step 1.3.2.2.2.1

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

$5 = - 3 a - 1$

$b = - 1$

$3 = a (- 2) + b$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

$5 = - 3 a - 1$

$b = - 1$

$3 = a (- 2) + b$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

$5 = - 3 a - 1$

$b = - 1$

$3 = a (- 2) + b$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $3 = a (- 2) + b$ with $- 1$ .

$3 = a (- 2) - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.4

Simplify $3 = a (- 2) - 1$ .

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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.

$3 = a (- 2) - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

$3 = a (- 2) - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.4.2

Simplify the right side.

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Step 1.3.2.4.2.1

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

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = a (- 1) + b$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.5

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

$1 = a (- 1) - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.6

Simplify $1 = a (- 1) - 1$ .

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Step 1.3.2.6.1

Simplify the left side.

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Step 1.3.2.6.1.1

Remove parentheses.

$1 = a (- 1) - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

$1 = a (- 1) - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.6.2

Simplify the right side.

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Step 1.3.2.6.2.1

Simplify each term.

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Step 1.3.2.6.2.1.1

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

$1 = - 1 a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.6.2.1.2

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

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = a + b$

$- 5 = a (2) + b$

Step 1.3.2.7

Replace all occurrences of $b$ in $- 3 = a + b$ with $- 1$ .

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 5 = a (2) + b$

Step 1.3.2.8

Simplify the left side.

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Step 1.3.2.8.1

Remove parentheses.

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 5 = a (2) + b$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 5 = a (2) + b$

Step 1.3.2.9

Replace all occurrences of $b$ in $- 5 = a (2) + b$ with $- 1$ .

$- 5 = a (2) - 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.2.10

Simplify $- 5 = a (2) - 1$ .

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Step 1.3.2.10.1

Simplify the left side.

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Step 1.3.2.10.1.1

Remove parentheses.

$- 5 = a (2) - 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 5 = a (2) - 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.2.10.2

Simplify the right side.

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Step 1.3.2.10.2.1

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

$- 5 = 2 a - 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 5 = 2 a - 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 5 = 2 a - 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 5 = 2 a - 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.3

Solve for $a$ in $- 5 = 2 a - 1$ .

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Step 1.3.3.1

Rewrite the equation as $2 a - 1 = - 5$ .

$2 a - 1 = - 5$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

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 $1$ to both sides of the equation.

$2 a = - 5 + 1$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.3.2.2

Add $- 5$ and $1$ .

$2 a = - 4$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$2 a = - 4$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.3.3

Divide each term in $2 a = - 4$ by $2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $2 a = - 4$ by $2$ .

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

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

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.

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

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

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

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

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

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

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

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Divide $- 4$ by $2$ .

$a = - 2$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$a = - 2$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$a = - 2$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$a = - 2$

$- 3 = a - 1$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4

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

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Step 1.3.4.1

Replace all occurrences of $a$ in $- 3 = a - 1$ with $- 2$ .

$- 3 = (- 2) - 1$

$a = - 2$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Subtract $1$ from $- 2$ .

$- 3 = - 3$

$a = - 2$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$- 3 = - 3$

$a = - 2$

$1 = - a - 1$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.3

Replace all occurrences of $a$ in $1 = - a - 1$ with $- 2$ .

$1 = - (- 2) - 1$

$- 3 = - 3$

$a = - 2$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $- (- 2) - 1$ .

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Step 1.3.4.4.1.1

Multiply $- 1$ by $- 2$ .

$1 = 2 - 1$

$- 3 = - 3$

$a = - 2$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.4.1.2

Subtract $1$ from $2$ .

$1 = 1$

$- 3 = - 3$

$a = - 2$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$3 = - 2 a - 1$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.5

Replace all occurrences of $a$ in $3 = - 2 a - 1$ with $- 2$ .

$3 = - 2 \cdot - 2 - 1$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify $- 2 \cdot - 2 - 1$ .

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Step 1.3.4.6.1.1

Multiply $- 2$ by $- 2$ .

$3 = 4 - 1$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.6.1.2

Subtract $1$ from $4$ .

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$5 = - 3 a - 1$

$b = - 1$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$5 = - 3 a - 1$

$b = - 1$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$5 = - 3 a - 1$

$b = - 1$

Step 1.3.4.7

Replace all occurrences of $a$ in $5 = - 3 a - 1$ with $- 2$ .

$5 = - 3 \cdot - 2 - 1$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$b = - 1$

Step 1.3.4.8

Simplify the right side.

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Step 1.3.4.8.1

Simplify $- 3 \cdot - 2 - 1$ .

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Step 1.3.4.8.1.1

Multiply $- 3$ by $- 2$ .

$5 = 6 - 1$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$b = - 1$

Step 1.3.4.8.1.2

Subtract $1$ from $6$ .

$5 = 5$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$b = - 1$

$5 = 5$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$b = - 1$

$5 = 5$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$b = - 1$

$5 = 5$

$3 = 3$

$1 = 1$

$- 3 = - 3$

$a = - 2$

$b = - 1$

Step 1.3.5

Remove any equations from the system that are always true.

$a = - 2$

$b = - 1$

Step 1.3.6

List all of the solutions.

$a = - 2, b = - 1$

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 $a = - 2$ , $b = - 1$ , and $x = - 3$ .

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Step 1.4.1.1

Multiply $- 2$ by $- 3$ .

$y = 6 - 1$

Step 1.4.1.2

Subtract $1$ from $6$ .

$y = 5$

Step 1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 3$ . This check passes since $y = 5$ and $y = 5$ .

$5 = 5$

Step 1.4.3

Calculate the value of $y$ when $a = - 2$ , $b = - 1$ , and $x = - 2$ .

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Step 1.4.3.1

Multiply $- 2$ by $- 2$ .

$y = 4 - 1$

Step 1.4.3.2

Subtract $1$ from $4$ .

$y = 3$

Step 1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 2$ . This check passes since $y = 3$ and $y = 3$ .

$3 = 3$

Step 1.4.5

Calculate the value of $y$ when $a = - 2$ , $b = - 1$ , and $x = - 1$ .

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Step 1.4.5.1

Multiply $- 2$ by $- 1$ .

$y = 2 - 1$

Step 1.4.5.2

Subtract $1$ from $2$ .

$y = 1$

Step 1.4.6

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check passes since $y = 1$ and $y = 1$ .

$1 = 1$

Step 1.4.7

Calculate the value of $y$ when $a = - 2$ , $b = - 1$ , and $x = 0$ .

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Step 1.4.7.1

Multiply $- 2$ by $0$ .

$y = 0 - 1$

Step 1.4.7.2

Subtract $1$ from $0$ .

$y = - 1$

Step 1.4.8

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = - 1$ and $y = - 1$ .

$- 1 = - 1$

Step 1.4.9

Calculate the value of $y$ when $a = - 2$ , $b = - 1$ , and $x = 1$ .

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Step 1.4.9.1

Multiply $- 2$ by $1$ .

$y = - 2 - 1$

Step 1.4.9.2

Subtract $1$ from $- 2$ .

$y = - 3$

Step 1.4.10

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = - 3$ and $y = - 3$ .

$- 3 = - 3$

Step 1.4.11

Calculate the value of $y$ when $a = - 2$ , $b = - 1$ , and $x = 2$ .

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Step 1.4.11.1

Multiply $- 2$ by $2$ .

$y = - 4 - 1$

Step 1.4.11.2

Subtract $1$ from $- 4$ .

$y = - 5$

Step 1.4.12

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = - 5$ and $y = - 5$ .

$- 5 = - 5$

Step 1.4.13

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 $y = - 2 x - 1$ .

$y = - 2 x - 1$