Find the Function Rule table[[x,y],[2,-6.6],[3,-9.9],[4,-13.2]]

$\begin{array}{c} x & y \\ 2 & - 6.6 \\ 3 & - 9.9 \\ 4 & - 13.2 \end{array}$

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 $y = a x + b$ .

$- 6.6 = a (2) + b - 9.9 = a (3) + b - 13.2 = a (4) + b$

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

Solve for $b$ in $- 6.6 = a (2) + b$ .

Rewrite the equation as $a (2) + b = - 6.6$ .

$a (2) + b = - 6.6$

$- 9.9 = a (3) + b$

$- 13.2 = a (4) + b$

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

$2 a + b = - 6.6$

$- 9.9 = a (3) + b$

$- 13.2 = a (4) + b$

Subtract $2 a$ from both sides of the equation.

$b = - 6.6 - 2 a$

$- 9.9 = a (3) + b$

$- 13.2 = a (4) + b$

$b = - 6.6 - 2 a$

$- 9.9 = a (3) + b$

$- 13.2 = a (4) + b$

Replace all occurrences of $b$ with $- 6.6 - 2 a$ in each equation.

Replace all occurrences of $b$ in $- 9.9 = a (3) + b$ with $- 6.6 - 2 a$ .

$- 9.9 = a (3) - 6.6 - 2 a$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

Simplify $- 9.9 = a (3) - 6.6 - 2 a$ .

Simplify the left side.

Remove parentheses.

$- 9.9 = a (3) - 6.6 - 2 a$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

$- 9.9 = a (3) - 6.6 - 2 a$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

Simplify the right side.

Simplify $a (3) - 6.6 - 2 a$ .

Move $3$ to the left of $a$ .

$- 9.9 = 3 a - 6.6 - 2 a$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

Subtract $2 a$ from $3 a$ .

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) + b$

Replace all occurrences of $b$ in $- 13.2 = a (4) + b$ with $- 6.6 - 2 a$ .

$- 13.2 = a (4) - 6.6 - 2 a$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Simplify $- 13.2 = a (4) - 6.6 - 2 a$ .

Simplify the left side.

Remove parentheses.

$- 13.2 = a (4) - 6.6 - 2 a$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = a (4) - 6.6 - 2 a$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Simplify the right side.

Simplify $a (4) - 6.6 - 2 a$ .

Move $4$ to the left of $a$ .

$- 13.2 = 4 a - 6.6 - 2 a$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Subtract $2 a$ from $4 a$ .

$- 13.2 = 2 a - 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = 2 a - 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = 2 a - 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = 2 a - 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$- 13.2 = 2 a - 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Solve for $a$ in $- 13.2 = 2 a - 6.6$ .

Rewrite the equation as $2 a - 6.6 = - 13.2$ .

$2 a - 6.6 = - 13.2$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

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

Add $6.6$ to both sides of the equation.

$2 a = - 13.2 + 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Add $- 13.2$ and $6.6$ .

$2 a = - 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$2 a = - 6.6$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Divide each term in $2 a = - 6.6$ by $2$ and simplify.

Divide each term in $2 a = - 6.6$ by $2$ .

$\frac{2 a}{2} = \frac{- 6.6}{2}$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{- 6.6}{2}$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Divide $a$ by $1$ .

$a = \frac{- 6.6}{2}$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$a = \frac{- 6.6}{2}$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$a = \frac{- 6.6}{2}$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Simplify the right side.

Divide $- 6.6$ by $2$ .

$a = - 3.3$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$a = - 3.3$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$a = - 3.3$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

$a = - 3.3$

$- 9.9 = a - 6.6$

$b = - 6.6 - 2 a$

Replace all occurrences of $a$ with $- 3.3$ in each equation.

Replace all occurrences of $a$ in $- 9.9 = a - 6.6$ with $- 3.3$ .

$- 9.9 = (- 3.3) - 6.6$

$a = - 3.3$

$b = - 6.6 - 2 a$

Simplify the right side.

Subtract $6.6$ from $- 3.3$ .

$- 9.9 = - 9.9$

$a = - 3.3$

$b = - 6.6 - 2 a$

$- 9.9 = - 9.9$

$a = - 3.3$

$b = - 6.6 - 2 a$

Replace all occurrences of $a$ in $b = - 6.6 - 2 a$ with $- 3.3$ .

$b = - 6.6 - 2 \cdot - 3.3$

$- 9.9 = - 9.9$

$a = - 3.3$

Simplify the right side.

Simplify $- 6.6 - 2 \cdot - 3.3$ .

Multiply $- 2$ by $- 3.3$ .

$b = - 6.6 + 6.6$

$- 9.9 = - 9.9$

$a = - 3.3$

Add $- 6.6$ and $6.6$ .

$b = 0$

$- 9.9 = - 9.9$

$a = - 3.3$

$b = 0$

$- 9.9 = - 9.9$

$a = - 3.3$

$b = 0$

$- 9.9 = - 9.9$

$a = - 3.3$

$b = 0$

$- 9.9 = - 9.9$

$a = - 3.3$

Remove any equations from the system that are always true.

$b = 0$

$a = - 3.3$

List all of the solutions.

$b = 0, a = - 3.3$

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 $a = - 3.3$ , $b = 0$ , and $x = 2$ .

Multiply $- 3.3$ by $2$ .

$y = - 6.6 + 0$

Add $- 6.6$ and $0$ .

$y = - 6.6$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = - 6.6$ and $y = - 6.6$ .

$- 6.6 = - 6.6$

Calculate the value of $y$ when $a = - 3.3$ , $b = 0$ , and $x = 3$ .

Multiply $- 3.3$ by $3$ .

$y = - 9.9 + 0$

Add $- 9.9$ and $0$ .

$y = - 9.9$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = - 9.9$ and $y = - 9.9$ .

$- 9.9 = - 9.9$

Calculate the value of $y$ when $a = - 3.3$ , $b = 0$ , and $x = 4$ .

Multiply $- 3.3$ by $4$ .

$y = - 13.2 + 0$

Add $- 13.2$ and $0$ .

$y = - 13.2$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 4$ . This check passes since $y = - 13.2$ and $y = - 13.2$ .

$- 13.2 = - 13.2$

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 $y = - 3.3 x$ .

$y = - 3.3 x$

Find the Function Rule table[[x,y],[2,-6.6],[3,-9.9],[4,-13.2]]

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