Find the Function Rule table[[x,y],[70,114.1],[100,118],[130,121.90]]

$\begin{array}{c} x & y \\ 70 & 114.1 \\ 100 & 118 \\ 130 & 121.9 \end{array}$

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 $y = a x + b$ .

$114.1 = a (70) + b 118 = a (100) + b 121.9 = a (130) + b$

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

Solve for $b$ in $114.1 = a (70) + b$ .

Rewrite the equation as $a (70) + b = 114.1$ .

$a (70) + b = 114.1$

$118 = a (100) + b$

$121.9 = a (130) + b$

Move $70$ to the left of $a$ .

$70 a + b = 114.1$

$118 = a (100) + b$

$121.9 = a (130) + b$

Subtract $70 a$ from both sides of the equation.

$b = 114.1 - 70 a$

$118 = a (100) + b$

$121.9 = a (130) + b$

$b = 114.1 - 70 a$

$118 = a (100) + b$

$121.9 = a (130) + b$

Replace all occurrences of $b$ with $114.1 - 70 a$ in each equation.

Replace all occurrences of $b$ in $118 = a (100) + b$ with $114.1 - 70 a$ .

$118 = a (100) + 114.1 - 70 a$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

Simplify $118 = a (100) + 114.1 - 70 a$ .

Simplify the left side.

Remove parentheses.

$118 = a (100) + 114.1 - 70 a$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

$118 = a (100) + 114.1 - 70 a$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

Simplify the right side.

Simplify $a (100) + 114.1 - 70 a$ .

Move $100$ to the left of $a$ .

$118 = 100 a + 114.1 - 70 a$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

Subtract $70 a$ from $100 a$ .

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = a (130) + b$

Replace all occurrences of $b$ in $121.9 = a (130) + b$ with $114.1 - 70 a$ .

$121.9 = a (130) + 114.1 - 70 a$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Simplify $121.9 = a (130) + 114.1 - 70 a$ .

Simplify the left side.

Remove parentheses.

$121.9 = a (130) + 114.1 - 70 a$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = a (130) + 114.1 - 70 a$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Simplify the right side.

Simplify $a (130) + 114.1 - 70 a$ .

Move $130$ to the left of $a$ .

$121.9 = 130 a + 114.1 - 70 a$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Subtract $70 a$ from $130 a$ .

$121.9 = 60 a + 114.1$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = 60 a + 114.1$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = 60 a + 114.1$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = 60 a + 114.1$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$121.9 = 60 a + 114.1$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Solve for $a$ in $121.9 = 60 a + 114.1$ .

Rewrite the equation as $60 a + 114.1 = 121.9$ .

$60 a + 114.1 = 121.9$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

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

Subtract $114.1$ from both sides of the equation.

$60 a = 121.9 - 114.1$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Subtract $114.1$ from $121.9$ .

$60 a = 7.8$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$60 a = 7.8$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Divide each term in $60 a = 7.8$ by $60$ and simplify.

Divide each term in $60 a = 7.8$ by $60$ .

$\frac{60 a}{60} = \frac{7.8}{60}$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Simplify the left side.

Cancel the common factor of $60$ .

Cancel the common factor.

$\frac{60 a}{60} = \frac{7.8}{60}$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Divide $a$ by $1$ .

$a = \frac{7.8}{60}$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$a = \frac{7.8}{60}$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$a = \frac{7.8}{60}$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Simplify the right side.

Divide $7.8$ by $60$ .

$a = 0.13$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$a = 0.13$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$a = 0.13$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

$a = 0.13$

$118 = 30 a + 114.1$

$b = 114.1 - 70 a$

Replace all occurrences of $a$ with $0.13$ in each equation.

Replace all occurrences of $a$ in $118 = 30 a + 114.1$ with $0.13$ .

$118 = 30 (0.13) + 114.1$

$a = 0.13$

$b = 114.1 - 70 a$

Simplify the right side.

Simplify $30 (0.13) + 114.1$ .

Multiply $30$ by $0.13$ .

$118 = 3.9 + 114.1$

$a = 0.13$

$b = 114.1 - 70 a$

Add $3.9$ and $114.1$ .

$118 = 118$

$a = 0.13$

$b = 114.1 - 70 a$

$118 = 118$

$a = 0.13$

$b = 114.1 - 70 a$

$118 = 118$

$a = 0.13$

$b = 114.1 - 70 a$

Replace all occurrences of $a$ in $b = 114.1 - 70 a$ with $0.13$ .

$b = 114.1 - 70 \cdot 0.13$

$118 = 118$

$a = 0.13$

Simplify the right side.

Simplify $114.1 - 70 \cdot 0.13$ .

Multiply $- 70$ by $0.13$ .

$b = 114.1 - 9.1$

$118 = 118$

$a = 0.13$

Subtract $9.1$ from $114.1$ .

$b = 105$

$118 = 118$

$a = 0.13$

$b = 105$

$118 = 118$

$a = 0.13$

$b = 105$

$118 = 118$

$a = 0.13$

$b = 105$

$118 = 118$

$a = 0.13$

Remove any equations from the system that are always true.

$b = 105$

$a = 0.13$

List all of the solutions.

$b = 105, a = 0.13$

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 $a = 0.13$ , $b = 105$ , and $x = 70$ .

Multiply $0.13$ by $70$ .

$y = 9.1 + 105$

Add $9.1$ and $105$ .

$y = 114.1$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 70$ . This check passes since $y = 114.1$ and $y = 114.1$ .

$114.1 = 114.1$

Calculate the value of $y$ when $a = 0.13$ , $b = 105$ , and $x = 100$ .

Multiply $0.13$ by $100$ .

$y = 13 + 105$

Add $13$ and $105$ .

$y = 118$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 100$ . This check passes since $y = 118$ and $y = 118$ .

$118 = 118$

Calculate the value of $y$ when $a = 0.13$ , $b = 105$ , and $x = 130$ .

Multiply $0.13$ by $130$ .

$y = 16.9 + 105$

Add $16.9$ and $105$ .

$y = 121.9$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 130$ . This check passes since $y = 121.9$ and $y = 121.9$ .

$121.9 = 121.9$

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 $y = 0.13 x + 105$ .

$y = 0.13 x + 105$

Find the Function Rule table[[x,y],[70,114.1],[100,118],[130,121.90]]

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