Find the Function Rule table[[x,y],[1,55],[2,90],[3,125]]

$\begin{array}{c} x & y \\ 1 & 55 \\ 2 & 90 \\ 3 & 125 \end{array}$

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 $y = a x + b$ .

$55 = a (1) + b 90 = a (2) + b 125 = a (3) + b$

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

Solve for $a$ in $55 = a + b$ .

Rewrite the equation as $a + b = 55$ .

$a + b = 55$

$90 = a (2) + b$

$125 = a (3) + b$

Subtract $b$ from both sides of the equation.

$a = 55 - b$

$90 = a (2) + b$

$125 = a (3) + b$

$a = 55 - b$

$90 = a (2) + b$

$125 = a (3) + b$

Replace all occurrences of $a$ with $55 - b$ in each equation.

Replace all occurrences of $a$ in $90 = a (2) + b$ with $55 - b$ .

$90 = (55 - b) (2) + b$

$a = 55 - b$

$125 = a (3) + b$

Simplify the right side.

Simplify $(55 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$90 = 55 \cdot 2 - b \cdot 2 + b$

$a = 55 - b$

$125 = a (3) + b$

Multiply $55$ by $2$ .

$90 = 110 - b \cdot 2 + b$

$a = 55 - b$

$125 = a (3) + b$

Multiply $2$ by $- 1$ .

$90 = 110 - 2 b + b$

$a = 55 - b$

$125 = a (3) + b$

$90 = 110 - 2 b + b$

$a = 55 - b$

$125 = a (3) + b$

Add $- 2 b$ and $b$ .

$90 = 110 - b$

$a = 55 - b$

$125 = a (3) + b$

$90 = 110 - b$

$a = 55 - b$

$125 = a (3) + b$

$90 = 110 - b$

$a = 55 - b$

$125 = a (3) + b$

Replace all occurrences of $a$ in $125 = a (3) + b$ with $55 - b$ .

$125 = (55 - b) (3) + b$

$90 = 110 - b$

$a = 55 - b$

Simplify the right side.

Simplify $(55 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$125 = 55 \cdot 3 - b \cdot 3 + b$

$90 = 110 - b$

$a = 55 - b$

Multiply $55$ by $3$ .

$125 = 165 - b \cdot 3 + b$

$90 = 110 - b$

$a = 55 - b$

Multiply $3$ by $- 1$ .

$125 = 165 - 3 b + b$

$90 = 110 - b$

$a = 55 - b$

$125 = 165 - 3 b + b$

$90 = 110 - b$

$a = 55 - b$

Add $- 3 b$ and $b$ .

$125 = 165 - 2 b$

$90 = 110 - b$

$a = 55 - b$

$125 = 165 - 2 b$

$90 = 110 - b$

$a = 55 - b$

$125 = 165 - 2 b$

$90 = 110 - b$

$a = 55 - b$

$125 = 165 - 2 b$

$90 = 110 - b$

$a = 55 - b$

Solve for $b$ in $125 = 165 - 2 b$ .

Rewrite the equation as $165 - 2 b = 125$ .

$165 - 2 b = 125$

$90 = 110 - b$

$a = 55 - b$

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

Subtract $165$ from both sides of the equation.

$- 2 b = 125 - 165$

$90 = 110 - b$

$a = 55 - b$

Subtract $165$ from $125$ .

$- 2 b = - 40$

$90 = 110 - b$

$a = 55 - b$

$- 2 b = - 40$

$90 = 110 - b$

$a = 55 - b$

Divide each term in $- 2 b = - 40$ by $- 2$ and simplify.

Divide each term in $- 2 b = - 40$ by $- 2$ .

$\frac{- 2 b}{- 2} = \frac{- 40}{- 2}$

$90 = 110 - b$

$a = 55 - b$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 b}{- 2} = \frac{- 40}{- 2}$

$90 = 110 - b$

$a = 55 - b$

Divide $b$ by $1$ .

$b = \frac{- 40}{- 2}$

$90 = 110 - b$

$a = 55 - b$

$b = \frac{- 40}{- 2}$

$90 = 110 - b$

$a = 55 - b$

$b = \frac{- 40}{- 2}$

$90 = 110 - b$

$a = 55 - b$

Simplify the right side.

Divide $- 40$ by $- 2$ .

$b = 20$

$90 = 110 - b$

$a = 55 - b$

$b = 20$

$90 = 110 - b$

$a = 55 - b$

$b = 20$

$90 = 110 - b$

$a = 55 - b$

$b = 20$

$90 = 110 - b$

$a = 55 - b$

Replace all occurrences of $b$ with $20$ in each equation.

Replace all occurrences of $b$ in $90 = 110 - b$ with $20$ .

$90 = 110 - (20)$

$b = 20$

$a = 55 - b$

Simplify the right side.

Simplify $110 - (20)$ .

Multiply $- 1$ by $20$ .

$90 = 110 - 20$

$b = 20$

$a = 55 - b$

Subtract $20$ from $110$ .

$90 = 90$

$b = 20$

$a = 55 - b$

$90 = 90$

$b = 20$

$a = 55 - b$

$90 = 90$

$b = 20$

$a = 55 - b$

Replace all occurrences of $b$ in $a = 55 - b$ with $20$ .

$a = 55 - (20)$

$90 = 90$

$b = 20$

Simplify the right side.

Simplify $55 - (20)$ .

Multiply $- 1$ by $20$ .

$a = 55 - 20$

$90 = 90$

$b = 20$

Subtract $20$ from $55$ .

$a = 35$

$90 = 90$

$b = 20$

$a = 35$

$90 = 90$

$b = 20$

$a = 35$

$90 = 90$

$b = 20$

$a = 35$

$90 = 90$

$b = 20$

Remove any equations from the system that are always true.

$a = 35$

$b = 20$

List all of the solutions.

$a = 35, b = 20$

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 $a = 35$ , $b = 20$ , and $x = 1$ .

Multiply $35$ by $1$ .

$y = 35 + 20$

Add $35$ and $20$ .

$y = 55$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 55$ and $y = 55$ .

$55 = 55$

Calculate the value of $y$ when $a = 35$ , $b = 20$ , and $x = 2$ .

Multiply $35$ by $2$ .

$y = 70 + 20$

Add $70$ and $20$ .

$y = 90$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 90$ and $y = 90$ .

$90 = 90$

Calculate the value of $y$ when $a = 35$ , $b = 20$ , and $x = 3$ .

Multiply $35$ by $3$ .

$y = 105 + 20$

Add $105$ and $20$ .

$y = 125$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 125$ and $y = 125$ .

$125 = 125$

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 $y = 35 x + 20$ .

$y = 35 x + 20$

Find the Function Rule table[[x,y],[1,55],[2,90],[3,125]]

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