Find the Function Rule table[[x,y],[1,5],[2,2],[3,-1],[4,-4],[5,-7]]

$\begin{array}{c} x & y \\ 1 & 5 \\ 2 & 2 \\ 3 & - 1 \\ 4 & - 4 \\ 5 & - 7 \end{array}$

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 $y = a x + b$ .

$5 = a (1) + b 2 = a (2) + b - 1 = a (3) + b - 4 = a (4) + b - 7 = a (5) + b$

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

Solve for $a$ in $5 = a + b$ .

Rewrite the equation as $a + b = 5$ .

$a + b = 5$

$2 = a (2) + b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Subtract $b$ from both sides of the equation.

$a = 5 - b$

$2 = a (2) + b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

$a = 5 - b$

$2 = a (2) + b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Replace all occurrences of $a$ with $5 - b$ in each equation.

Replace all occurrences of $a$ in $2 = a (2) + b$ with $5 - b$ .

$2 = (5 - b) (2) + b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Simplify the right side.

Simplify $(5 - b) (2) + b$ .

Simplify each term.

Apply the distributive property.

$2 = 5 \cdot 2 - b \cdot 2 + b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Multiply $5$ by $2$ .

$2 = 10 - b \cdot 2 + b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Multiply $2$ by $- 1$ .

$2 = 10 - 2 b + b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

$2 = 10 - 2 b + b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Add $- 2 b$ and $b$ .

$2 = 10 - b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

$2 = 10 - b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

$2 = 10 - b$

$a = 5 - b$

$- 1 = a (3) + b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Replace all occurrences of $a$ in $- 1 = a (3) + b$ with $5 - b$ .

$- 1 = (5 - b) (3) + b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Simplify the right side.

Simplify $(5 - b) (3) + b$ .

Simplify each term.

Apply the distributive property.

$- 1 = 5 \cdot 3 - b \cdot 3 + b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Multiply $5$ by $3$ .

$- 1 = 15 - b \cdot 3 + b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Multiply $3$ by $- 1$ .

$- 1 = 15 - 3 b + b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

$- 1 = 15 - 3 b + b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Add $- 3 b$ and $b$ .

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = a (4) + b$

$- 7 = a (5) + b$

Replace all occurrences of $a$ in $- 4 = a (4) + b$ with $5 - b$ .

$- 4 = (5 - b) (4) + b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

Simplify the right side.

Simplify $(5 - b) (4) + b$ .

Simplify each term.

Apply the distributive property.

$- 4 = 5 \cdot 4 - b \cdot 4 + b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

Multiply $5$ by $4$ .

$- 4 = 20 - b \cdot 4 + b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

Multiply $4$ by $- 1$ .

$- 4 = 20 - 4 b + b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

$- 4 = 20 - 4 b + b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

Add $- 4 b$ and $b$ .

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = a (5) + b$

Replace all occurrences of $a$ in $- 7 = a (5) + b$ with $5 - b$ .

$- 7 = (5 - b) (5) + b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Simplify the right side.

Simplify $(5 - b) (5) + b$ .

Simplify each term.

Apply the distributive property.

$- 7 = 5 \cdot 5 - b \cdot 5 + b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Multiply $5$ by $5$ .

$- 7 = 25 - b \cdot 5 + b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Multiply $5$ by $- 1$ .

$- 7 = 25 - 5 b + b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = 25 - 5 b + b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Add $- 5 b$ and $b$ .

$- 7 = 25 - 4 b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = 25 - 4 b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = 25 - 4 b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 7 = 25 - 4 b$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Solve for $b$ in $- 7 = 25 - 4 b$ .

Rewrite the equation as $25 - 4 b = - 7$ .

$25 - 4 b = - 7$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

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

Subtract $25$ from both sides of the equation.

$- 4 b = - 7 - 25$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Subtract $25$ from $- 7$ .

$- 4 b = - 32$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 4 b = - 32$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Divide each term in $- 4 b = - 32$ by $- 4$ and simplify.

Divide each term in $- 4 b = - 32$ by $- 4$ .

$\frac{- 4 b}{- 4} = \frac{- 32}{- 4}$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

$\frac{- 4 b}{- 4} = \frac{- 32}{- 4}$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Divide $b$ by $1$ .

$b = \frac{- 32}{- 4}$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$b = \frac{- 32}{- 4}$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$b = \frac{- 32}{- 4}$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Simplify the right side.

Divide $- 32$ by $- 4$ .

$b = 8$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$b = 8$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$b = 8$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$b = 8$

$- 4 = 20 - 3 b$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

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

Replace all occurrences of $b$ in $- 4 = 20 - 3 b$ with $8$ .

$- 4 = 20 - 3 \cdot 8$

$b = 8$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Simplify the right side.

Simplify $20 - 3 \cdot 8$ .

Multiply $- 3$ by $8$ .

$- 4 = 20 - 24$

$b = 8$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Subtract $24$ from $20$ .

$- 4 = - 4$

$b = 8$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = - 4$

$b = 8$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

$- 4 = - 4$

$b = 8$

$- 1 = 15 - 2 b$

$2 = 10 - b$

$a = 5 - b$

Replace all occurrences of $b$ in $- 1 = 15 - 2 b$ with $8$ .

$- 1 = 15 - 2 \cdot 8$

$- 4 = - 4$

$b = 8$

$2 = 10 - b$

$a = 5 - b$

Simplify the right side.

Simplify $15 - 2 \cdot 8$ .

Multiply $- 2$ by $8$ .

$- 1 = 15 - 16$

$- 4 = - 4$

$b = 8$

$2 = 10 - b$

$a = 5 - b$

Subtract $16$ from $15$ .

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$2 = 10 - b$

$a = 5 - b$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$2 = 10 - b$

$a = 5 - b$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$2 = 10 - b$

$a = 5 - b$

Replace all occurrences of $b$ in $2 = 10 - b$ with $8$ .

$2 = 10 - (8)$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = 5 - b$

Simplify the right side.

Simplify $10 - (8)$ .

Multiply $- 1$ by $8$ .

$2 = 10 - 8$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = 5 - b$

Subtract $8$ from $10$ .

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = 5 - b$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = 5 - b$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = 5 - b$

Replace all occurrences of $b$ in $a = 5 - b$ with $8$ .

$a = 5 - (8)$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

Simplify the right side.

Simplify $5 - (8)$ .

Multiply $- 1$ by $8$ .

$a = 5 - 8$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

Subtract $8$ from $5$ .

$a = - 3$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = - 3$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = - 3$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

$a = - 3$

$2 = 2$

$- 1 = - 1$

$- 4 = - 4$

$b = 8$

Remove any equations from the system that are always true.

$a = - 3$

$b = 8$

List all of the solutions.

$a = - 3, b = 8$

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 $a = - 3$ , $b = 8$ , and $x = 1$ .

Multiply $- 3$ by $1$ .

$y = - 3 + 8$

Add $- 3$ and $8$ .

$y = 5$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 1$ . This check passes since $y = 5$ and $y = 5$ .

$5 = 5$

Calculate the value of $y$ when $a = - 3$ , $b = 8$ , and $x = 2$ .

Multiply $- 3$ by $2$ .

$y = - 6 + 8$

Add $- 6$ and $8$ .

$y = 2$

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

$2 = 2$

Calculate the value of $y$ when $a = - 3$ , $b = 8$ , and $x = 3$ .

Multiply $- 3$ by $3$ .

$y = - 9 + 8$

Add $- 9$ and $8$ .

$y = - 1$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = - 1$ and $y = - 1$ .

$- 1 = - 1$

Calculate the value of $y$ when $a = - 3$ , $b = 8$ , and $x = 4$ .

Multiply $- 3$ by $4$ .

$y = - 12 + 8$

Add $- 12$ and $8$ .

$y = - 4$

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

$- 4 = - 4$

Calculate the value of $y$ when $a = - 3$ , $b = 8$ , and $x = 5$ .

Multiply $- 3$ by $5$ .

$y = - 15 + 8$

Add $- 15$ and $8$ .

$y = - 7$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 5$ . This check passes since $y = - 7$ and $y = - 7$ .

$- 7 = - 7$

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 $y = - 3 x + 8$ .

$y = - 3 x + 8$

Find the Function Rule table[[x,y],[1,5],[2,2],[3,-1],[4,-4],[5,-7]]

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