Find the Function Rule table[[x,y],[6,9.6],[14,22.4],[22,35.2]]

$\begin{array}{c} x & y \\ 6 & 9.6 \\ 14 & 22.4 \\ 22 & 35.2 \end{array}$

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 $y = a x + b$ .

$9.6 = a (6) + b 22.4 = a (14) + b 35.2 = a (22) + b$

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

Solve for $b$ in $9.6 = a (6) + b$ .

Rewrite the equation as $a (6) + b = 9.6$ .

$a (6) + b = 9.6$

$22.4 = a (14) + b$

$35.2 = a (22) + b$

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

$6 a + b = 9.6$

$22.4 = a (14) + b$

$35.2 = a (22) + b$

Subtract $6 a$ from both sides of the equation.

$b = 9.6 - 6 a$

$22.4 = a (14) + b$

$35.2 = a (22) + b$

$b = 9.6 - 6 a$

$22.4 = a (14) + b$

$35.2 = a (22) + b$

Replace all occurrences of $b$ with $9.6 - 6 a$ in each equation.

Replace all occurrences of $b$ in $22.4 = a (14) + b$ with $9.6 - 6 a$ .

$22.4 = a (14) + 9.6 - 6 a$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

Simplify $22.4 = a (14) + 9.6 - 6 a$ .

Simplify the left side.

Remove parentheses.

$22.4 = a (14) + 9.6 - 6 a$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

$22.4 = a (14) + 9.6 - 6 a$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

Simplify the right side.

Simplify $a (14) + 9.6 - 6 a$ .

Move $14$ to the left of $a$ .

$22.4 = 14 a + 9.6 - 6 a$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

Subtract $6 a$ from $14 a$ .

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = a (22) + b$

Replace all occurrences of $b$ in $35.2 = a (22) + b$ with $9.6 - 6 a$ .

$35.2 = a (22) + 9.6 - 6 a$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Simplify $35.2 = a (22) + 9.6 - 6 a$ .

Simplify the left side.

Remove parentheses.

$35.2 = a (22) + 9.6 - 6 a$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = a (22) + 9.6 - 6 a$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Simplify the right side.

Simplify $a (22) + 9.6 - 6 a$ .

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

$35.2 = 22 a + 9.6 - 6 a$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Subtract $6 a$ from $22 a$ .

$35.2 = 16 a + 9.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = 16 a + 9.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = 16 a + 9.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = 16 a + 9.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$35.2 = 16 a + 9.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Solve for $a$ in $35.2 = 16 a + 9.6$ .

Rewrite the equation as $16 a + 9.6 = 35.2$ .

$16 a + 9.6 = 35.2$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

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

Subtract $9.6$ from both sides of the equation.

$16 a = 35.2 - 9.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Subtract $9.6$ from $35.2$ .

$16 a = 25.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$16 a = 25.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Divide each term in $16 a = 25.6$ by $16$ and simplify.

Divide each term in $16 a = 25.6$ by $16$ .

$\frac{16 a}{16} = \frac{25.6}{16}$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 a}{16} = \frac{25.6}{16}$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Divide $a$ by $1$ .

$a = \frac{25.6}{16}$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$a = \frac{25.6}{16}$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$a = \frac{25.6}{16}$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Simplify the right side.

Divide $25.6$ by $16$ .

$a = 1.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$a = 1.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$a = 1.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

$a = 1.6$

$22.4 = 8 a + 9.6$

$b = 9.6 - 6 a$

Replace all occurrences of $a$ with $1.6$ in each equation.

Replace all occurrences of $a$ in $22.4 = 8 a + 9.6$ with $1.6$ .

$22.4 = 8 (1.6) + 9.6$

$a = 1.6$

$b = 9.6 - 6 a$

Simplify the right side.

Simplify $8 (1.6) + 9.6$ .

Multiply $8$ by $1.6$ .

$22.4 = 12.8 + 9.6$

$a = 1.6$

$b = 9.6 - 6 a$

Add $12.8$ and $9.6$ .

$22.4 = 22.4$

$a = 1.6$

$b = 9.6 - 6 a$

$22.4 = 22.4$

$a = 1.6$

$b = 9.6 - 6 a$

$22.4 = 22.4$

$a = 1.6$

$b = 9.6 - 6 a$

Replace all occurrences of $a$ in $b = 9.6 - 6 a$ with $1.6$ .

$b = 9.6 - 6 \cdot 1.6$

$22.4 = 22.4$

$a = 1.6$

Simplify the right side.

Simplify $9.6 - 6 \cdot 1.6$ .

Multiply $- 6$ by $1.6$ .

$b = 9.6 - 9.6$

$22.4 = 22.4$

$a = 1.6$

Subtract $9.6$ from $9.6$ .

$b = 0$

$22.4 = 22.4$

$a = 1.6$

$b = 0$

$22.4 = 22.4$

$a = 1.6$

$b = 0$

$22.4 = 22.4$

$a = 1.6$

$b = 0$

$22.4 = 22.4$

$a = 1.6$

Remove any equations from the system that are always true.

$b = 0$

$a = 1.6$

List all of the solutions.

$b = 0, a = 1.6$

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 $a = 1.6$ , $b = 0$ , and $x = 6$ .

Multiply $1.6$ by $6$ .

$y = 9.6 + 0$

Add $9.6$ and $0$ .

$y = 9.6$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 6$ . This check passes since $y = 9.6$ and $y = 9.6$ .

$9.6 = 9.6$

Calculate the value of $y$ when $a = 1.6$ , $b = 0$ , and $x = 14$ .

Multiply $1.6$ by $14$ .

$y = 22.4 + 0$

Add $22.4$ and $0$ .

$y = 22.4$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 14$ . This check passes since $y = 22.4$ and $y = 22.4$ .

$22.4 = 22.4$

Calculate the value of $y$ when $a = 1.6$ , $b = 0$ , and $x = 22$ .

Multiply $1.6$ by $22$ .

$y = 35.2 + 0$

Add $35.2$ and $0$ .

$y = 35.2$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 22$ . This check passes since $y = 35.2$ and $y = 35.2$ .

$35.2 = 35.2$

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 $y = 1.6 x$ .

$y = 1.6 x$

Find the Function Rule table[[x,y],[6,9.6],[14,22.4],[22,35.2]]

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