Find the Function Rule table[[x,y],[0,0],[1,5],[2,10],[3,15]]

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 5 \\ 2 & 10 \\ 3 & 15 \end{array}$

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 $y = a x + b$ .

$0 = a (0) + b 5 = a (1) + b 10 = a (2) + b 15 = a (3) + b$

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

Rewrite the equation as $b = 0$ .

$b = 0$

$5 = a + b$

$10 = a (2) + b$

$15 = a (3) + b$

Replace all occurrences of $b$ with $0$ in each equation.

Replace all occurrences of $b$ in $5 = a + b$ with $0$ .

$5 = a + 0$

$b = 0$

$10 = a (2) + b$

$15 = a (3) + b$

Simplify $5 = a + 0$ .

Simplify the left side.

Remove parentheses.

$5 = a + 0$

$b = 0$

$10 = a (2) + b$

$15 = a (3) + b$

$5 = a + 0$

$b = 0$

$10 = a (2) + b$

$15 = a (3) + b$

Simplify the right side.

Add $a$ and $0$ .

$5 = a$

$b = 0$

$10 = a (2) + b$

$15 = a (3) + b$

$5 = a$

$b = 0$

$10 = a (2) + b$

$15 = a (3) + b$

$5 = a$

$b = 0$

$10 = a (2) + b$

$15 = a (3) + b$

Replace all occurrences of $b$ in $10 = a (2) + b$ with $0$ .

$10 = a (2) + 0$

$5 = a$

$b = 0$

$15 = a (3) + b$

Simplify $10 = a (2) + 0$ .

Simplify the left side.

Remove parentheses.

$10 = a (2) + 0$

$5 = a$

$b = 0$

$15 = a (3) + b$

$10 = a (2) + 0$

$5 = a$

$b = 0$

$15 = a (3) + b$

Simplify the right side.

Simplify $a (2) + 0$ .

Add $a (2)$ and $0$ .

$10 = a (2)$

$5 = a$

$b = 0$

$15 = a (3) + b$

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

$10 = 2 a$

$5 = a$

$b = 0$

$15 = a (3) + b$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = a (3) + b$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = a (3) + b$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = a (3) + b$

Replace all occurrences of $b$ in $15 = a (3) + b$ with $0$ .

$15 = a (3) + 0$

$10 = 2 a$

$5 = a$

$b = 0$

Simplify $15 = a (3) + 0$ .

Simplify the left side.

Remove parentheses.

$15 = a (3) + 0$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = a (3) + 0$

$10 = 2 a$

$5 = a$

$b = 0$

Simplify the right side.

Simplify $a (3) + 0$ .

Add $a (3)$ and $0$ .

$15 = a (3)$

$10 = 2 a$

$5 = a$

$b = 0$

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

$15 = 3 a$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = 3 a$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = 3 a$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = 3 a$

$10 = 2 a$

$5 = a$

$b = 0$

$15 = 3 a$

$10 = 2 a$

$5 = a$

$b = 0$

Solve for $a$ in $15 = 3 a$ .

Rewrite the equation as $3 a = 15$ .

$3 a = 15$

$10 = 2 a$

$5 = a$

$b = 0$

Divide each term in $3 a = 15$ by $3$ and simplify.

Divide each term in $3 a = 15$ by $3$ .

$\frac{3 a}{3} = \frac{15}{3}$

$10 = 2 a$

$5 = a$

$b = 0$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 a}{3} = \frac{15}{3}$

$10 = 2 a$

$5 = a$

$b = 0$

Divide $a$ by $1$ .

$a = \frac{15}{3}$

$10 = 2 a$

$5 = a$

$b = 0$

$a = \frac{15}{3}$

$10 = 2 a$

$5 = a$

$b = 0$

$a = \frac{15}{3}$

$10 = 2 a$

$5 = a$

$b = 0$

Simplify the right side.

Divide $15$ by $3$ .

$a = 5$

$10 = 2 a$

$5 = a$

$b = 0$

$a = 5$

$10 = 2 a$

$5 = a$

$b = 0$

$a = 5$

$10 = 2 a$

$5 = a$

$b = 0$

$a = 5$

$10 = 2 a$

$5 = a$

$b = 0$

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

Replace all occurrences of $a$ in $10 = 2 a$ with $5$ .

$10 = 2 (5)$

$a = 5$

$5 = a$

$b = 0$

Simplify the right side.

Multiply $2$ by $5$ .

$10 = 10$

$a = 5$

$5 = a$

$b = 0$

$10 = 10$

$a = 5$

$5 = a$

$b = 0$

$10 = 10$

$a = 5$

$5 = a$

$b = 0$

Remove any equations from the system that are always true.

$a = 5$

$5 = a$

$b = 0$

List all of the solutions.

$a = 5, b = 0$

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 $a = 5$ , $b = 0$ , and $x = 0$ .

Multiply $5$ by $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Calculate the value of $y$ when $a = 5$ , $b = 0$ , and $x = 1$ .

Multiply $5$ by $1$ .

$y = 5 + 0$

Add $5$ and $0$ .

$y = 5$

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 $a = 5$ , $b = 0$ , and $x = 2$ .

Multiply $5$ by $2$ .

$y = 10 + 0$

Add $10$ and $0$ .

$y = 10$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 10$ and $y = 10$ .

$10 = 10$

Calculate the value of $y$ when $a = 5$ , $b = 0$ , and $x = 3$ .

Multiply $5$ by $3$ .

$y = 15 + 0$

Add $15$ and $0$ .

$y = 15$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 15$ and $y = 15$ .

$15 = 15$

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 $y = 5 x$ .

$y = 5 x$

Find the Function Rule table[[x,y],[0,0],[1,5],[2,10],[3,15]]

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