Find the Function Rule table[[x,y],[125,75],[150,85],[175,95]]

$\begin{array}{c} x & y \\ 125 & 75 \\ 150 & 85 \\ 175 & 95 \end{array}$

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 $y = a x + b$ .

$75 = a (125) + b 85 = a (150) + b 95 = a (175) + b$

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

Solve for $b$ in $75 = a (125) + b$ .

Rewrite the equation as $a (125) + b = 75$ .

$a (125) + b = 75$

$85 = a (150) + b$

$95 = a (175) + b$

Move $125$ to the left of $a$ .

$125 a + b = 75$

$85 = a (150) + b$

$95 = a (175) + b$

Subtract $125 a$ from both sides of the equation.

$b = 75 - 125 a$

$85 = a (150) + b$

$95 = a (175) + b$

$b = 75 - 125 a$

$85 = a (150) + b$

$95 = a (175) + b$

Replace all occurrences of $b$ with $75 - 125 a$ in each equation.

Replace all occurrences of $b$ in $85 = a (150) + b$ with $75 - 125 a$ .

$85 = a (150) + 75 - 125 a$

$b = 75 - 125 a$

$95 = a (175) + b$

Simplify $85 = a (150) + 75 - 125 a$ .

Simplify the left side.

Remove parentheses.

$85 = a (150) + 75 - 125 a$

$b = 75 - 125 a$

$95 = a (175) + b$

$85 = a (150) + 75 - 125 a$

$b = 75 - 125 a$

$95 = a (175) + b$

Simplify the right side.

Simplify $a (150) + 75 - 125 a$ .

Move $150$ to the left of $a$ .

$85 = 150 a + 75 - 125 a$

$b = 75 - 125 a$

$95 = a (175) + b$

Subtract $125 a$ from $150 a$ .

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = a (175) + b$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = a (175) + b$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = a (175) + b$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = a (175) + b$

Replace all occurrences of $b$ in $95 = a (175) + b$ with $75 - 125 a$ .

$95 = a (175) + 75 - 125 a$

$85 = 25 a + 75$

$b = 75 - 125 a$

Simplify $95 = a (175) + 75 - 125 a$ .

Simplify the left side.

Remove parentheses.

$95 = a (175) + 75 - 125 a$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = a (175) + 75 - 125 a$

$85 = 25 a + 75$

$b = 75 - 125 a$

Simplify the right side.

Simplify $a (175) + 75 - 125 a$ .

Move $175$ to the left of $a$ .

$95 = 175 a + 75 - 125 a$

$85 = 25 a + 75$

$b = 75 - 125 a$

Subtract $125 a$ from $175 a$ .

$95 = 50 a + 75$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = 50 a + 75$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = 50 a + 75$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = 50 a + 75$

$85 = 25 a + 75$

$b = 75 - 125 a$

$95 = 50 a + 75$

$85 = 25 a + 75$

$b = 75 - 125 a$

Solve for $a$ in $95 = 50 a + 75$ .

Rewrite the equation as $50 a + 75 = 95$ .

$50 a + 75 = 95$

$85 = 25 a + 75$

$b = 75 - 125 a$

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

Subtract $75$ from both sides of the equation.

$50 a = 95 - 75$

$85 = 25 a + 75$

$b = 75 - 125 a$

Subtract $75$ from $95$ .

$50 a = 20$

$85 = 25 a + 75$

$b = 75 - 125 a$

$50 a = 20$

$85 = 25 a + 75$

$b = 75 - 125 a$

Divide each term in $50 a = 20$ by $50$ and simplify.

Divide each term in $50 a = 20$ by $50$ .

$\frac{50 a}{50} = \frac{20}{50}$

$85 = 25 a + 75$

$b = 75 - 125 a$

Simplify the left side.

Cancel the common factor of $50$ .

Cancel the common factor.

$\frac{50 a}{50} = \frac{20}{50}$

$85 = 25 a + 75$

$b = 75 - 125 a$

Divide $a$ by $1$ .

$a = \frac{20}{50}$

$85 = 25 a + 75$

$b = 75 - 125 a$

$a = \frac{20}{50}$

$85 = 25 a + 75$

$b = 75 - 125 a$

$a = \frac{20}{50}$

$85 = 25 a + 75$

$b = 75 - 125 a$

Simplify the right side.

Cancel the common factor of $20$ and $50$ .

Factor $10$ out of $20$ .

$a = \frac{10 (2)}{50}$

$85 = 25 a + 75$

$b = 75 - 125 a$

Cancel the common factors.

Factor $10$ out of $50$ .

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

$85 = 25 a + 75$

$b = 75 - 125 a$

Cancel the common factor.

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

$85 = 25 a + 75$

$b = 75 - 125 a$

Rewrite the expression.

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

$85 = 25 a + 75$

$b = 75 - 125 a$

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

$85 = 25 a + 75$

$b = 75 - 125 a$

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

$85 = 25 a + 75$

$b = 75 - 125 a$

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

$85 = 25 a + 75$

$b = 75 - 125 a$

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

$85 = 25 a + 75$

$b = 75 - 125 a$

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

$85 = 25 a + 75$

$b = 75 - 125 a$

Replace all occurrences of $a$ with $\frac{2}{5}$ in each equation.

Replace all occurrences of $a$ in $85 = 25 a + 75$ with $\frac{2}{5}$ .

$85 = 25 (\frac{2}{5}) + 75$

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

$b = 75 - 125 a$

Simplify the right side.

Simplify $25 (\frac{2}{5}) + 75$ .

Simplify each term.

Cancel the common factor of $5$ .

Factor $5$ out of $25$ .

$85 = 5 (5) (\frac{2}{5}) + 75$

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

$b = 75 - 125 a$

Cancel the common factor.

$85 = 5 \cdot (5 (\frac{2}{5})) + 75$

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

$b = 75 - 125 a$

Rewrite the expression.

$85 = 5 \cdot 2 + 75$

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

$b = 75 - 125 a$

$85 = 5 \cdot 2 + 75$

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

$b = 75 - 125 a$

Multiply $5$ by $2$ .

$85 = 10 + 75$

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

$b = 75 - 125 a$

$85 = 10 + 75$

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

$b = 75 - 125 a$

Add $10$ and $75$ .

$85 = 85$

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

$b = 75 - 125 a$

$85 = 85$

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

$b = 75 - 125 a$

$85 = 85$

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

$b = 75 - 125 a$

Replace all occurrences of $a$ in $b = 75 - 125 a$ with $\frac{2}{5}$ .

$b = 75 - 125 (\frac{2}{5})$

$85 = 85$

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

Simplify the right side.

Simplify $75 - 125 (\frac{2}{5})$ .

Simplify each term.

Cancel the common factor of $5$ .

Factor $5$ out of $- 125$ .

$b = 75 + 5 (- 25) (\frac{2}{5})$

$85 = 85$

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

Cancel the common factor.

$b = 75 + 5 \cdot (- 25 (\frac{2}{5}))$

$85 = 85$

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

Rewrite the expression.

$b = 75 - 25 \cdot 2$

$85 = 85$

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

$b = 75 - 25 \cdot 2$

$85 = 85$

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

Multiply $- 25$ by $2$ .

$b = 75 - 50$

$85 = 85$

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

$b = 75 - 50$

$85 = 85$

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

Subtract $50$ from $75$ .

$b = 25$

$85 = 85$

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

$b = 25$

$85 = 85$

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

$b = 25$

$85 = 85$

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

$b = 25$

$85 = 85$

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

Remove any equations from the system that are always true.

$b = 25$

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

List all of the solutions.

$b = 25, a = \frac{2}{5}$

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 $a = \frac{2}{5}$ , $b = 25$ , and $x = 125$ .

Simplify each term.

Cancel the common factor of $5$ .

Factor $5$ out of $125$ .

$y = \frac{2}{5} \cdot (5 (25)) + 25$

Cancel the common factor.

$y = \frac{2}{5} \cdot (5 \cdot 25) + 25$

Rewrite the expression.

$y = 2 \cdot 25 + 25$

Multiply $2$ by $25$ .

$y = 50 + 25$

Add $50$ and $25$ .

$y = 75$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 125$ . This check passes since $y = 75$ and $y = 75$ .

$75 = 75$

Calculate the value of $y$ when $a = \frac{2}{5}$ , $b = 25$ , and $x = 150$ .

Simplify each term.

Cancel the common factor of $5$ .

Factor $5$ out of $150$ .

$y = \frac{2}{5} \cdot (5 (30)) + 25$

Cancel the common factor.

$y = \frac{2}{5} \cdot (5 \cdot 30) + 25$

Rewrite the expression.

$y = 2 \cdot 30 + 25$

Multiply $2$ by $30$ .

$y = 60 + 25$

Add $60$ and $25$ .

$y = 85$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 150$ . This check passes since $y = 85$ and $y = 85$ .

$85 = 85$

Calculate the value of $y$ when $a = \frac{2}{5}$ , $b = 25$ , and $x = 175$ .

Simplify each term.

Cancel the common factor of $5$ .

Factor $5$ out of $175$ .

$y = \frac{2}{5} \cdot (5 (35)) + 25$

Cancel the common factor.

$y = \frac{2}{5} \cdot (5 \cdot 35) + 25$

Rewrite the expression.

$y = 2 \cdot 35 + 25$

Multiply $2$ by $35$ .

$y = 70 + 25$

Add $70$ and $25$ .

$y = 95$

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 175$ . This check passes since $y = 95$ and $y = 95$ .

$95 = 95$

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 $y = \frac{2 x}{5} + 25$ .

$y = \frac{2 x}{5} + 25$

Find the Function Rule table[[x,y],[125,75],[150,85],[175,95]]

Related Questions

Support LCMGCF.com by disabling your adblocker.