Find the Function Rule table[[x,y],[8,31.6],[15,54],[22,76.4]]

$\begin{array}{c} x & y \\ 8 & 31.6 \\ 15 & 54 \\ 22 & 76.4 \end{array}$

Step 1

Check if the function rule is linear.

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Build a set of equations from the table such that $y = a x + b$ .

$31.6 = a (8) + b 54 = a (15) + b 76.4 = a (22) + b$

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

Solve for $b$ in $31.6 = a (8) + b$ .

Rewrite the equation as $a (8) + b = 31.6$ .

$a (8) + b = 31.6$

$54 = a (15) + b$

$76.4 = a (22) + b$

Move $8$ to the left of $a$ .

$8 a + b = 31.6$

$54 = a (15) + b$

$76.4 = a (22) + b$

Subtract $8 a$ from both sides of the equation.

$b = 31.6 - 8 a$

$54 = a (15) + b$

$76.4 = a (22) + b$

$b = 31.6 - 8 a$

$54 = a (15) + b$

$76.4 = a (22) + b$

Replace all occurrences of $b$ with $31.6 - 8 a$ in each equation.

Replace all occurrences of $b$ in $54 = a (15) + b$ with $31.6 - 8 a$ .

$54 = a (15) + 31.6 - 8 a$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

Simplify $54 = a (15) + 31.6 - 8 a$ .

Simplify the left side.

Remove parentheses.

$54 = a (15) + 31.6 - 8 a$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

$54 = a (15) + 31.6 - 8 a$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

Simplify the right side.

Simplify $a (15) + 31.6 - 8 a$ .

Move $15$ to the left of $a$ .

$54 = 15 a + 31.6 - 8 a$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

Subtract $8 a$ from $15 a$ .

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = a (22) + b$

Replace all occurrences of $b$ in $76.4 = a (22) + b$ with $31.6 - 8 a$ .

$76.4 = a (22) + 31.6 - 8 a$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Simplify $76.4 = a (22) + 31.6 - 8 a$ .

Simplify the left side.

Remove parentheses.

$76.4 = a (22) + 31.6 - 8 a$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = a (22) + 31.6 - 8 a$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Simplify the right side.

Simplify $a (22) + 31.6 - 8 a$ .

Move $22$ to the left of $a$ .

$76.4 = 22 a + 31.6 - 8 a$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Subtract $8 a$ from $22 a$ .

$76.4 = 14 a + 31.6$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = 14 a + 31.6$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = 14 a + 31.6$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = 14 a + 31.6$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$76.4 = 14 a + 31.6$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Solve for $a$ in $76.4 = 14 a + 31.6$ .

Rewrite the equation as $14 a + 31.6 = 76.4$ .

$14 a + 31.6 = 76.4$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

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

Subtract $31.6$ from both sides of the equation.

$14 a = 76.4 - 31.6$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Subtract $31.6$ from $76.4$ .

$14 a = 44.8$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$14 a = 44.8$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Divide each term in $14 a = 44.8$ by $14$ and simplify.

Divide each term in $14 a = 44.8$ by $14$ .

$\frac{14 a}{14} = \frac{44.8}{14}$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Simplify the left side.

Cancel the common factor of $14$ .

Cancel the common factor.

$\frac{14 a}{14} = \frac{44.8}{14}$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Divide $a$ by $1$ .

$a = \frac{44.8}{14}$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$a = \frac{44.8}{14}$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$a = \frac{44.8}{14}$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Simplify the right side.

Divide $44.8$ by $14$ .

$a = 3.2$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$a = 3.2$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$a = 3.2$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

$a = 3.2$

$54 = 7 a + 31.6$

$b = 31.6 - 8 a$

Replace all occurrences of $a$ with $3.2$ in each equation.

Replace all occurrences of $a$ in $54 = 7 a + 31.6$ with $3.2$ .

$54 = 7 (3.2) + 31.6$

$a = 3.2$

$b = 31.6 - 8 a$

Simplify the right side.

Simplify $7 (3.2) + 31.6$ .

Multiply $7$ by $3.2$ .

$54 = 22.4 + 31.6$

$a = 3.2$

$b = 31.6 - 8 a$

Add $22.4$ and $31.6$ .

$54 = 54$

$a = 3.2$

$b = 31.6 - 8 a$

$54 = 54$

$a = 3.2$

$b = 31.6 - 8 a$

$54 = 54$

$a = 3.2$

$b = 31.6 - 8 a$

Replace all occurrences of $a$ in $b = 31.6 - 8 a$ with $3.2$ .

$b = 31.6 - 8 \cdot 3.2$

$54 = 54$

$a = 3.2$

Simplify the right side.

Simplify $31.6 - 8 \cdot 3.2$ .

Multiply $- 8$ by $3.2$ .

$b = 31.6 - 25.6$

$54 = 54$

$a = 3.2$

Subtract $25.6$ from $31.6$ .

$b = 6$

$54 = 54$

$a = 3.2$

$b = 6$

$54 = 54$

$a = 3.2$

$b = 6$

$54 = 54$

$a = 3.2$

$b = 6$

$54 = 54$

$a = 3.2$

Remove any equations from the system that are always true.

$b = 6$

$a = 3.2$

List all of the solutions.

$b = 6, a = 3.2$

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

Calculate the value of $y$ when $a = 3.2$ , $b = 6$ , and $x = 8$ .

Multiply $3.2$ by $8$ .

$y = 25.6 + 6$

Add $25.6$ and $6$ .

$y = 31.6$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 31.6$ and $y = 31.6$ .

$31.6 = 31.6$

Calculate the value of $y$ when $a = 3.2$ , $b = 6$ , and $x = 15$ .

Multiply $3.2$ by $15$ .

$y = 48 + 6$

Add $48$ and $6$ .

$y = 54$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 15$ . This check passes since $y = 54$ and $y = 54$ .

$54 = 54$

Calculate the value of $y$ when $a = 3.2$ , $b = 6$ , and $x = 22$ .

Multiply $3.2$ by $22$ .

$y = 70.4 + 6$

Add $70.4$ and $6$ .

$y = 76.4$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 22$ . This check passes since $y = 76.4$ and $y = 76.4$ .

$76.4 = 76.4$

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 $y = 3.2 x + 6$ .

$y = 3.2 x + 6$

Find the Function Rule table[[x,y],[8,31.6],[15,54],[22,76.4]]

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