Pre-Algebra Examples

$\begin{array}{c} x & y \\ - 1 & 8 \\ 0 & 4 \\ 1 & 0 \\ 2 & - 4 \end{array}$

Step 1

Check if the function rule is linear.

Tap for more steps...

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 $y = a x + b$ .

$8 = a (- 1) + b 4 = a (0) + b 0 = a (1) + b - 4 = a (2) + b$

Step 1.3

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

Tap for more steps...

Step 1.3.1

Rewrite the equation as $b = 4$ .

$b = 4$

$8 = a (- 1) + b$

$0 = a + b$

$- 4 = a (2) + b$

Step 1.3.2

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

Tap for more steps...

Step 1.3.2.1

Replace all occurrences of $b$ in $8 = a (- 1) + b$ with $4$ .

$8 = a (- 1) + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

Step 1.3.2.2

Simplify $8 = a (- 1) + 4$ .

Tap for more steps...

Step 1.3.2.2.1

Simplify the left side.

Tap for more steps...

Step 1.3.2.2.1.1

Remove parentheses.

$8 = a (- 1) + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

$8 = a (- 1) + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

Step 1.3.2.2.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.2.2.1

Simplify each term.

Tap for more steps...

Step 1.3.2.2.2.1.1

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

$8 = - 1 a + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

Step 1.3.2.2.2.1.2

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

$8 = - a + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

$8 = - a + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

$8 = - a + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

$8 = - a + 4$

$b = 4$

$0 = a + b$

$- 4 = a (2) + b$

Step 1.3.2.3

Replace all occurrences of $b$ in $0 = a + b$ with $4$ .

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$- 4 = a (2) + b$

Step 1.3.2.4

Simplify the left side.

Tap for more steps...

Step 1.3.2.4.1

Remove parentheses.

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$- 4 = a (2) + b$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$- 4 = a (2) + b$

Step 1.3.2.5

Replace all occurrences of $b$ in $- 4 = a (2) + b$ with $4$ .

$- 4 = a (2) + 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.2.6

Simplify $- 4 = a (2) + 4$ .

Tap for more steps...

Step 1.3.2.6.1

Simplify the left side.

Tap for more steps...

Step 1.3.2.6.1.1

Remove parentheses.

$- 4 = a (2) + 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$- 4 = a (2) + 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.2.6.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.6.2.1

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

$- 4 = 2 a + 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$- 4 = 2 a + 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$- 4 = 2 a + 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$- 4 = 2 a + 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.3

Solve for $a$ in $- 4 = 2 a + 4$ .

Tap for more steps...

Step 1.3.3.1

Rewrite the equation as $2 a + 4 = - 4$ .

$2 a + 4 = - 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.3.2

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

Tap for more steps...

Step 1.3.3.2.1

Subtract $4$ from both sides of the equation.

$2 a = - 4 - 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.3.2.2

Subtract $4$ from $- 4$ .

$2 a = - 8$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$2 a = - 8$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.3.3

Divide each term in $2 a = - 8$ by $2$ and simplify.

Tap for more steps...

Step 1.3.3.3.1

Divide each term in $2 a = - 8$ by $2$ .

$\frac{2 a}{2} = \frac{- 8}{2}$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.3.3.2

Simplify the left side.

Tap for more steps...

Step 1.3.3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 8}{2}$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 8}{2}$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$a = \frac{- 8}{2}$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$a = \frac{- 8}{2}$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.3.3.3

Simplify the right side.

Tap for more steps...

Step 1.3.3.3.3.1

Divide $- 8$ by $2$ .

$a = - 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$a = - 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$a = - 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

$a = - 4$

$0 = a + 4$

$8 = - a + 4$

$b = 4$

Step 1.3.4

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

Tap for more steps...

Step 1.3.4.1

Replace all occurrences of $a$ in $0 = a + 4$ with $- 4$ .

$0 = (- 4) + 4$

$a = - 4$

$8 = - a + 4$

$b = 4$

Step 1.3.4.2

Simplify the right side.

Tap for more steps...

Step 1.3.4.2.1

Add $- 4$ and $4$ .

$0 = 0$

$a = - 4$

$8 = - a + 4$

$b = 4$

$0 = 0$

$a = - 4$

$8 = - a + 4$

$b = 4$

Step 1.3.4.3

Replace all occurrences of $a$ in $8 = - a + 4$ with $- 4$ .

$8 = - (- 4) + 4$

$0 = 0$

$a = - 4$

$b = 4$

Step 1.3.4.4

Simplify the right side.

Tap for more steps...

Step 1.3.4.4.1

Simplify $- (- 4) + 4$ .

Tap for more steps...

Step 1.3.4.4.1.1

Multiply $- 1$ by $- 4$ .

$8 = 4 + 4$

$0 = 0$

$a = - 4$

$b = 4$

Step 1.3.4.4.1.2

Add $4$ and $4$ .

$8 = 8$

$0 = 0$

$a = - 4$

$b = 4$

$8 = 8$

$0 = 0$

$a = - 4$

$b = 4$

$8 = 8$

$0 = 0$

$a = - 4$

$b = 4$

$8 = 8$

$0 = 0$

$a = - 4$

$b = 4$

Step 1.3.5

Remove any equations from the system that are always true.

$a = - 4$

$b = 4$

Step 1.3.6

List all of the solutions.

$a = - 4, b = 4$

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.

Tap for more steps...

Step 1.4.1

Calculate the value of $y$ when $a = - 4$ , $b = 4$ , and $x = - 1$ .

Tap for more steps...

Step 1.4.1.1

Multiply $- 4$ by $- 1$ .

$y = 4 + 4$

Step 1.4.1.2

Add $4$ and $4$ .

$y = 8$

Step 1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check passes since $y = 8$ and $y = 8$ .

$8 = 8$

Step 1.4.3

Calculate the value of $y$ when $a = - 4$ , $b = 4$ , and $x = 0$ .

Tap for more steps...

Step 1.4.3.1

Multiply $- 4$ by $0$ .

$y = 0 + 4$

Step 1.4.3.2

Add $0$ and $4$ .

$y = 4$

Step 1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 4$ and $y = 4$ .

$4 = 4$

Step 1.4.5

Calculate the value of $y$ when $a = - 4$ , $b = 4$ , and $x = 1$ .

Tap for more steps...

Step 1.4.5.1

Multiply $- 4$ by $1$ .

$y = - 4 + 4$

Step 1.4.5.2

Add $- 4$ and $4$ .

$y = 0$

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 $y = 0$ and $y = 0$ .

$0 = 0$

Step 1.4.7

Calculate the value of $y$ when $a = - 4$ , $b = 4$ , and $x = 2$ .

Tap for more steps...

Step 1.4.7.1

Multiply $- 4$ by $2$ .

$y = - 8 + 4$

Step 1.4.7.2

Add $- 8$ and $4$ .

$y = - 4$

Step 1.4.8

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = - 4$ and $y = - 4$ .

$- 4 = - 4$

Step 1.4.9

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 $y = - 4 x + 4$ .

$y = - 4 x + 4$