Find the Function Rule table[[x,y],[11,23.8],[13,25.4],[15,27]]

$\begin{array}{c} x & y \\ 11 & 23.8 \\ 13 & 25.4 \\ 15 & 27 \end{array}$

Check if the function rule is linear.

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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 $y = a x + b$ .

$23.8 = a (11) + b 25.4 = a (13) + b 27 = a (15) + b$

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

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Simplify each equation.

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Move $11$ to the left of $a$ .

$23.8 = 11 a + b$

$25.4 = a (13) + b$

$27 = a (15) + b$

Move $13$ to the left of $a$ .

$23.8 = 11 a + b$

$25.4 = 13 a + b$

$27 = a (15) + b$

Move $15$ to the left of $a$ .

$23.8 = 11 a + b$

$25.4 = 13 a + b$

$27 = 15 a + b$

$23.8 = 11 a + b$

$25.4 = 13 a + b$

$27 = 15 a + b$

Solve for $b$ in the first equation.

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Rewrite the equation as $11 a + b = 23.8$ .

$11 a + b = 23.8$

$25.4 = 13 a + b$

$27 = 15 a + b$

Subtract $11 a$ from both sides of the equation.

$b = 23.8 - 11 a$

$25.4 = 13 a + b$

$27 = 15 a + b$

$b = 23.8 - 11 a$

$25.4 = 13 a + b$

$27 = 15 a + b$

Replace all occurrences of $b$ with $23.8 - 11 a$ in each equation.

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Replace all occurrences of $b$ in $25.4 = 13 a + b$ with $23.8 - 11 a$ .

$b = 23.8 - 11 a$

$25.4 = 13 a + 23.8 - 11 a$

$27 = 15 a + b$

Replace all occurrences of $b$ in $27 = 15 a + b$ with $23.8 - 11 a$ .

$b = 23.8 - 11 a$

$25.4 = 13 a + 23.8 - 11 a$

$27 = 15 a + 23.8 - 11 a$

$b = 23.8 - 11 a$

$25.4 = 13 a + 23.8 - 11 a$

$27 = 15 a + 23.8 - 11 a$

Simplify each equation.

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Remove parentheses.

$b = 23.8 - 11 a$

$25.4 = 13 a + 23.8 - 11 a$

$27 = 15 a + 23.8 - 11 a$

Subtract $11 a$ from $13 a$ .

$b = 23.8 - 11 a$

$25.4 = 2 a + 23.8$

$27 = 15 a + 23.8 - 11 a$

Remove parentheses.

$b = 23.8 - 11 a$

$25.4 = 2 a + 23.8$

$27 = 15 a + 23.8 - 11 a$

Subtract $11 a$ from $15 a$ .

$b = 23.8 - 11 a$

$25.4 = 2 a + 23.8$

$27 = 4 a + 23.8$

$b = 23.8 - 11 a$

$25.4 = 2 a + 23.8$

$27 = 4 a + 23.8$

Solve for $a$ in the second equation.

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Rewrite the equation as $2 a + 23.8 = 25.4$ .

$b = 23.8 - 11 a$

$2 a + 23.8 = 25.4$

$27 = 4 a + 23.8$

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

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Subtract $23.8$ from both sides of the equation.

$b = 23.8 - 11 a$

$2 a = 25.4 - 23.8$

$27 = 4 a + 23.8$

Subtract $23.8$ from $25.4$ .

$b = 23.8 - 11 a$

$2 a = 1.6$

$27 = 4 a + 23.8$

$b = 23.8 - 11 a$

$2 a = 1.6$

$27 = 4 a + 23.8$

Divide each term by $2$ and simplify.

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Divide each term in $2 a = 1.6$ by $2$ .

$b = 23.8 - 11 a$

$\frac{2 a}{2} = \frac{1.6}{2}$

$27 = 4 a + 23.8$

Cancel the common factor of $2$ .

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Cancel the common factor.

$b = 23.8 - 11 a$

$\frac{2 a}{2} = \frac{1.6}{2}$

$27 = 4 a + 23.8$

Divide $a$ by $1$ .

$b = 23.8 - 11 a$

$a = \frac{1.6}{2}$

$27 = 4 a + 23.8$

$b = 23.8 - 11 a$

$a = \frac{1.6}{2}$

$27 = 4 a + 23.8$

Divide $1.6$ by $2$ .

$b = 23.8 - 11 a$

$a = 0.8$

$27 = 4 a + 23.8$

$b = 23.8 - 11 a$

$a = 0.8$

$27 = 4 a + 23.8$

$b = 23.8 - 11 a$

$a = 0.8$

$27 = 4 a + 23.8$

Replace all occurrences of $a$ with $0.8$ in each equation.

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Replace all occurrences of $a$ in $b = 23.8 - 11 a$ with $0.8$ .

$b = 23.8 - 11 \cdot 0.8$

$a = 0.8$

$27 = 4 a + 23.8$

Replace all occurrences of $a$ in $27 = 4 a + 23.8$ with $0.8$ .

$b = 23.8 - 11 \cdot 0.8$

$a = 0.8$

$27 = 4 (0.8) + 23.8$

$b = 23.8 - 11 \cdot 0.8$

$a = 0.8$

$27 = 4 (0.8) + 23.8$

Simplify.

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Simplify $23.8 - 11 \cdot 0.8$ .

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Multiply $- 11$ by $0.8$ .

$b = 23.8 - 8.8$

$a = 0.8$

$27 = 4 (0.8) + 23.8$

Subtract $8.8$ from $23.8$ .

$b = 15$

$a = 0.8$

$27 = 4 (0.8) + 23.8$

$b = 15$

$a = 0.8$

$27 = 4 (0.8) + 23.8$

Simplify $4 (0.8) + 23.8$ .

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Multiply $4$ by $0.8$ .

$b = 15$

$a = 0.8$

$27 = 3.2 + 23.8$

Add $3.2$ and $23.8$ .

$b = 15$

$a = 0.8$

$27 = 27$

$b = 15$

$a = 0.8$

$27 = 27$

$b = 15$

$a = 0.8$

$27 = 27$

Since $27 = 27$ , the equation will always be true.

$b = 15$

$a = 0.8$

Always true

Remove any equations from the system that are always true.

$b = 15$

$a = 0.8$

List the solutions to the system of equations.

$b = 15$

$a = 0.8$

Always true

List all of the solutions.

$b = 15 a = 0.8$

Calculate the value of $y$ using each $x$ value in the relation and compare this value to the given $y$ value in the relation.

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Calculate the value of $y$ when $a = 0.8$ , $b = 15$ , and $x = 11$ .

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Multiply $0.8$ by $11$ .

$y = 8.8 + 15$

Add $8.8$ and $15$ .

$y = 23.8$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 11$ . This check passes since $y = 23.8$ and $y = 23.8$ .

$23.8 = 23.8$

Calculate the value of $y$ when $a = 0.8$ , $b = 15$ , and $x = 13$ .

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Multiply $0.8$ by $13$ .

$y = 10.4 + 15$

Add $10.4$ and $15$ .

$y = 25.4$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 13$ . This check passes since $y = 25.4$ and $y = 25.4$ .

$25.4 = 25.4$

Calculate the value of $y$ when $a = 0.8$ , $b = 15$ , and $x = 15$ .

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Multiply $0.8$ by $15$ .

$y = 12 + 15$

Add $12$ and $15$ .

$y = 27$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 15$ . This check passes since $y = 27$ and $y = 27$ .

$27 = 27$

Since $y = y$ for the corresponding $x$ values, the function is linear.

The function is linear

Since all $y = y$ , the function is linear and follows the form $y = 0.8 x + 15$ .

$y = 0.8 x + 15$

Find the Function Rule table[[x,y],[11,23.8],[13,25.4],[15,27]]

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