Find the Function Rule table[[x,y],[2,6.50],[8,5.00],[14,3.50]]

$\begin{array}{c} x & y \\ 2 & 6.5 \\ 8 & 5 \\ 14 & 3.5 \end{array}$

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 $y = a x + b$ .

$6.5 = a (2) + b 5 = a (8) + b 3.5 = a (14) + b$

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

Solve for $b$ in $6.5 = a (2) + b$ .

Rewrite the equation as $a (2) + b = 6.5$ .

$a (2) + b = 6.5$

$5 = a (8) + b$

$3.5 = a (14) + b$

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

$2 a + b = 6.5$

$5 = a (8) + b$

$3.5 = a (14) + b$

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

$b = 6.5 - 2 a$

$5 = a (8) + b$

$3.5 = a (14) + b$

$b = 6.5 - 2 a$

$5 = a (8) + b$

$3.5 = a (14) + b$

Replace all occurrences of $b$ with $6.5 - 2 a$ in each equation.

Replace all occurrences of $b$ in $5 = a (8) + b$ with $6.5 - 2 a$ .

$5 = a (8) + 6.5 - 2 a$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

Simplify $5 = a (8) + 6.5 - 2 a$ .

Simplify the left side.

Remove parentheses.

$5 = a (8) + 6.5 - 2 a$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

$5 = a (8) + 6.5 - 2 a$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

Simplify the right side.

Simplify $a (8) + 6.5 - 2 a$ .

Move $8$ to the left of $a$ .

$5 = 8 a + 6.5 - 2 a$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

Subtract $2 a$ from $8 a$ .

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = a (14) + b$

Replace all occurrences of $b$ in $3.5 = a (14) + b$ with $6.5 - 2 a$ .

$3.5 = a (14) + 6.5 - 2 a$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Simplify $3.5 = a (14) + 6.5 - 2 a$ .

Simplify the left side.

Remove parentheses.

$3.5 = a (14) + 6.5 - 2 a$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = a (14) + 6.5 - 2 a$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Simplify the right side.

Simplify $a (14) + 6.5 - 2 a$ .

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

$3.5 = 14 a + 6.5 - 2 a$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Subtract $2 a$ from $14 a$ .

$3.5 = 12 a + 6.5$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = 12 a + 6.5$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = 12 a + 6.5$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = 12 a + 6.5$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$3.5 = 12 a + 6.5$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Solve for $a$ in $3.5 = 12 a + 6.5$ .

Rewrite the equation as $12 a + 6.5 = 3.5$ .

$12 a + 6.5 = 3.5$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

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

Subtract $6.5$ from both sides of the equation.

$12 a = 3.5 - 6.5$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Subtract $6.5$ from $3.5$ .

$12 a = - 3$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$12 a = - 3$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Divide each term in $12 a = - 3$ by $12$ and simplify.

Divide each term in $12 a = - 3$ by $12$ .

$\frac{12 a}{12} = \frac{- 3}{12}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 a}{12} = \frac{- 3}{12}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Divide $a$ by $1$ .

$a = \frac{- 3}{12}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$a = \frac{- 3}{12}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$a = \frac{- 3}{12}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Simplify the right side.

Cancel the common factor of $- 3$ and $12$ .

Factor $3$ out of $- 3$ .

$a = \frac{3 (- 1)}{12}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Cancel the common factors.

Factor $3$ out of $12$ .

$a = \frac{3 \cdot - 1}{3 \cdot 4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Cancel the common factor.

$a = \frac{3 \cdot - 1}{3 \cdot 4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Rewrite the expression.

$a = \frac{- 1}{4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$a = \frac{- 1}{4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$a = \frac{- 1}{4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Move the negative in front of the fraction.

$a = - \frac{1}{4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$a = - \frac{1}{4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$a = - \frac{1}{4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

$a = - \frac{1}{4}$

$5 = 6 a + 6.5$

$b = 6.5 - 2 a$

Replace all occurrences of $a$ with $- \frac{1}{4}$ in each equation.

Replace all occurrences of $a$ in $5 = 6 a + 6.5$ with $- \frac{1}{4}$ .

$5 = 6 (- \frac{1}{4}) + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Simplify the right side.

Simplify $6 (- \frac{1}{4}) + 6.5$ .

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$5 = 6 (\frac{- 1}{4}) + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Factor $2$ out of $6$ .

$5 = 2 (3) (\frac{- 1}{4}) + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Factor $2$ out of $4$ .

$5 = 2 \cdot (3 (\frac{- 1}{2 \cdot 2})) + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Cancel the common factor.

$5 = 2 \cdot (3 (\frac{- 1}{2 \cdot 2})) + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Rewrite the expression.

$5 = 3 (\frac{- 1}{2}) + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

$5 = 3 (\frac{- 1}{2}) + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Combine $3$ and $\frac{- 1}{2}$ .

$5 = \frac{3 \cdot - 1}{2} + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Multiply $3$ by $- 1$ .

$5 = \frac{- 3}{2} + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Move the negative in front of the fraction.

$5 = - \frac{3}{2} + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

$5 = - \frac{3}{2} + 6.5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

To write $6.5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5 = - \frac{3}{2} + 6.5 \cdot \frac{2}{2}$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Combine $6.5$ and $\frac{2}{2}$ .

$5 = - \frac{3}{2} + \frac{6.5 \cdot 2}{2}$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Combine the numerators over the common denominator.

$5 = \frac{- 3 + 6.5 \cdot 2}{2}$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Simplify the numerator.

Multiply $6.5$ by $2$ .

$5 = \frac{- 3 + 13}{2}$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Add $- 3$ and $13$ .

$5 = \frac{10}{2}$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

$5 = \frac{10}{2}$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Divide $10$ by $2$ .

$5 = 5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

$5 = 5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

$5 = 5$

$a = - \frac{1}{4}$

$b = 6.5 - 2 a$

Replace all occurrences of $a$ in $b = 6.5 - 2 a$ with $- \frac{1}{4}$ .

$b = 6.5 - 2 (- \frac{1}{4})$

$5 = 5$

$a = - \frac{1}{4}$

Simplify the right side.

Simplify $6.5 - 2 (- \frac{1}{4})$ .

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$b = 6.5 - 2 (\frac{- 1}{4})$

$5 = 5$

$a = - \frac{1}{4}$

Factor $2$ out of $- 2$ .

$b = 6.5 + 2 (- 1) (\frac{- 1}{4})$

$5 = 5$

$a = - \frac{1}{4}$

Factor $2$ out of $4$ .

$b = 6.5 + 2 \cdot (- 1 \frac{- 1}{2 \cdot 2})$

$5 = 5$

$a = - \frac{1}{4}$

Cancel the common factor.

$b = 6.5 + 2 \cdot (- 1 \frac{- 1}{2 \cdot 2})$

$5 = 5$

$a = - \frac{1}{4}$

Rewrite the expression.

$b = 6.5 - 1 (\frac{- 1}{2})$

$5 = 5$

$a = - \frac{1}{4}$

$b = 6.5 - 1 (\frac{- 1}{2})$

$5 = 5$

$a = - \frac{1}{4}$

Move the negative in front of the fraction.

$b = 6.5 - 1 (- \frac{1}{2})$

$5 = 5$

$a = - \frac{1}{4}$

Multiply $- 1 (- \frac{1}{2})$ .

Multiply $- 1$ by $- 1$ .

$b = 6.5 + 1 (\frac{1}{2})$

$5 = 5$

$a = - \frac{1}{4}$

Multiply $\frac{1}{2}$ by $1$ .

$b = 6.5 + \frac{1}{2}$

$5 = 5$

$a = - \frac{1}{4}$

$b = 6.5 + \frac{1}{2}$

$5 = 5$

$a = - \frac{1}{4}$

$b = 6.5 + \frac{1}{2}$

$5 = 5$

$a = - \frac{1}{4}$

To write $6.5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = 6.5 \cdot \frac{2}{2} + \frac{1}{2}$

$5 = 5$

$a = - \frac{1}{4}$

Combine $6.5$ and $\frac{2}{2}$ .

$b = \frac{6.5 \cdot 2}{2} + \frac{1}{2}$

$5 = 5$

$a = - \frac{1}{4}$

Combine the numerators over the common denominator.

$b = \frac{6.5 \cdot 2 + 1}{2}$

$5 = 5$

$a = - \frac{1}{4}$

Simplify the numerator.

Multiply $6.5$ by $2$ .

$b = \frac{13 + 1}{2}$

$5 = 5$

$a = - \frac{1}{4}$

Add $13$ and $1$ .

$b = \frac{14}{2}$

$5 = 5$

$a = - \frac{1}{4}$

$b = \frac{14}{2}$

$5 = 5$

$a = - \frac{1}{4}$

Divide $14$ by $2$ .

$b = 7$

$5 = 5$

$a = - \frac{1}{4}$

$b = 7$

$5 = 5$

$a = - \frac{1}{4}$

$b = 7$

$5 = 5$

$a = - \frac{1}{4}$

$b = 7$

$5 = 5$

$a = - \frac{1}{4}$

Remove any equations from the system that are always true.

$b = 7$

$a = - \frac{1}{4}$

List all of the solutions.

$b = 7, a = - \frac{1}{4}$

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 $a = - \frac{1}{4}$ , $b = 7$ , and $x = 2$ .

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$y = \frac{- 1}{4} \cdot 2 + 7$

Factor $2$ out of $4$ .

$y = \frac{- 1}{2 (2)} \cdot 2 + 7$

Cancel the common factor.

$y = \frac{- 1}{2 \cdot 2} \cdot 2 + 7$

Rewrite the expression.

$y = \frac{- 1}{2} + 7$

Move the negative in front of the fraction.

$y = - \frac{1}{2} + 7$

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{1}{2} + 7 \cdot \frac{2}{2}$

Combine $7$ and $\frac{2}{2}$ .

$y = - \frac{1}{2} + \frac{7 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 1 + 7 \cdot 2}{2}$

Simplify the numerator.

Multiply $7$ by $2$ .

$y = \frac{- 1 + 14}{2}$

Add $- 1$ and $14$ .

$y = \frac{13}{2}$

Cancel the common factor of $13$ and $2$ .

Rewrite $13$ as $1 (13)$ .

$y = \frac{1 (13)}{2}$

Cancel the common factors.

Rewrite $2$ as $1 (2)$ .

$y = \frac{1 \cdot 13}{1 \cdot 2}$

Cancel the common factor.

$y = \frac{1 \cdot 13}{1 \cdot 2}$

Rewrite the expression.

$y = \frac{13}{2}$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = \frac{13}{2}$ and $y = 6.5$ .

$\frac{13}{2} = 6.5$

Calculate the value of $y$ when $a = - \frac{1}{4}$ , $b = 7$ , and $x = 8$ .

Simplify each term.

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$y = \frac{- 1}{4} \cdot 8 + 7$

Factor $4$ out of $8$ .

$y = \frac{- 1}{4} \cdot (4 (2)) + 7$

Cancel the common factor.

$y = \frac{- 1}{4} \cdot (4 \cdot 2) + 7$

Rewrite the expression.

$y = - 1 \cdot 2 + 7$

Multiply $- 1$ by $2$ .

$y = - 2 + 7$

Add $- 2$ and $7$ .

$y = 5$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 5$ and $y = 5$ .

$5 = 5$

Calculate the value of $y$ when $a = - \frac{1}{4}$ , $b = 7$ , and $x = 14$ .

Simplify each term.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$y = \frac{- 1}{4} \cdot 14 + 7$

Factor $2$ out of $4$ .

$y = \frac{- 1}{2 (2)} \cdot 14 + 7$

Factor $2$ out of $14$ .

$y = \frac{- 1}{2 \cdot 2} \cdot (2 \cdot 7) + 7$

Cancel the common factor.

$y = \frac{- 1}{2 \cdot 2} \cdot (2 \cdot 7) + 7$

Rewrite the expression.

$y = \frac{- 1}{2} \cdot 7 + 7$

Combine $\frac{- 1}{2}$ and $7$ .

$y = \frac{- 1 \cdot 7}{2} + 7$

Multiply $- 1$ by $7$ .

$y = \frac{- 7}{2} + 7$

Move the negative in front of the fraction.

$y = - \frac{7}{2} + 7$

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{7}{2} + 7 \cdot \frac{2}{2}$

Combine $7$ and $\frac{2}{2}$ .

$y = - \frac{7}{2} + \frac{7 \cdot 2}{2}$

Combine the numerators over the common denominator.

$y = \frac{- 7 + 7 \cdot 2}{2}$

Simplify the numerator.

Multiply $7$ by $2$ .

$y = \frac{- 7 + 14}{2}$

Add $- 7$ and $14$ .

$y = \frac{7}{2}$

Cancel the common factor of $7$ and $2$ .

Rewrite $7$ as $1 (7)$ .

$y = \frac{1 (7)}{2}$

Cancel the common factors.

Rewrite $2$ as $1 (2)$ .

$y = \frac{1 \cdot 7}{1 \cdot 2}$

Cancel the common factor.

$y = \frac{1 \cdot 7}{1 \cdot 2}$

Rewrite the expression.

$y = \frac{7}{2}$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 14$ . This check passes since $y = \frac{7}{2}$ and $y = 3.5$ .

$\frac{7}{2} = 3.5$

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 $y = - \frac{x}{4} + 7$ .

$y = - \frac{x}{4} + 7$

Find the Function Rule table[[x,y],[2,6.50],[8,5.00],[14,3.50]]

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