Algebra Examples

$f (- 3) = 5$ and $f (6) = - 2$

Step 1

$f (- 3) = 5$ , which means $(- 3, 5)$ is a point on the line. $f (6) = - 2$ , which means $(6, - 2)$ is a point on the line, too.

$(- 3, 5), (6, - 2)$

Step 2

Find the slope of the line between $(- 3, 5)$ and $(6, - 2)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 2.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 2.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 2.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{- 2 - (5)}{6 - (- 3)}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Multiply $- 1$ by $5$ .

$m = \frac{- 2 - 5}{6 - (- 3)}$

Step 2.4.1.2

Subtract $5$ from $- 2$ .

$m = \frac{- 7}{6 - (- 3)}$

Step 2.4.2

Simplify the denominator.

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Step 2.4.2.1

Multiply $- 1$ by $- 3$ .

$m = \frac{- 7}{6 + 3}$

Step 2.4.2.2

Add $6$ and $3$ .

$m = \frac{- 7}{9}$

Step 2.4.3

Move the negative in front of the fraction.

$m = - \frac{7}{9}$

Step 3

Use the slope $- \frac{7}{9}$ and a given point $(- 3, 5)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - \frac{7}{9} \cdot (x - (- 3))$

Step 4

Simplify the equation and keep it in point-slope form.

$y - 5 = - \frac{7}{9} \cdot (x + 3)$

Step 5

Solve for $y$ .

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Step 5.1

Simplify $- \frac{7}{9} \cdot (x + 3)$ .

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Step 5.1.1

Rewrite.

$y - 5 = 0 + 0 - \frac{7}{9} \cdot (x + 3)$

Step 5.1.2

Simplify terms.

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Step 5.1.2.1

Apply the distributive property.

$y - 5 = - \frac{7}{9} x - \frac{7}{9} \cdot 3$

Step 5.1.2.2

Combine $x$ and $\frac{7}{9}$ .

$y - 5 = - \frac{x \cdot 7}{9} - \frac{7}{9} \cdot 3$

Step 5.1.2.3

Cancel the common factor of $3$ .

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Step 5.1.2.3.1

Move the leading negative in $- \frac{7}{9}$ into the numerator.

$y - 5 = - \frac{x \cdot 7}{9} + \frac{- 7}{9} \cdot 3$

Step 5.1.2.3.2

Factor $3$ out of $9$ .

$y - 5 = - \frac{x \cdot 7}{9} + \frac{- 7}{3 (3)} \cdot 3$

Step 5.1.2.3.3

Cancel the common factor.

$y - 5 = - \frac{x \cdot 7}{9} + \frac{- 7}{3 \cdot 3} \cdot 3$

Step 5.1.2.3.4

Rewrite the expression.

$y - 5 = - \frac{x \cdot 7}{9} + \frac{- 7}{3}$

Step 5.1.3

Simplify each term.

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Step 5.1.3.1

Move $7$ to the left of $x$ .

$y - 5 = - \frac{7 \cdot x}{9} + \frac{- 7}{3}$

Step 5.1.3.2

Move the negative in front of the fraction.

$y - 5 = - \frac{7 x}{9} - \frac{7}{3}$

Step 5.2

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

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Step 5.2.1

Add $5$ to both sides of the equation.

$y = - \frac{7 x}{9} - \frac{7}{3} + 5$

Step 5.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = - \frac{7 x}{9} - \frac{7}{3} + 5 \cdot \frac{3}{3}$

Step 5.2.3

Combine $5$ and $\frac{3}{3}$ .

$y = - \frac{7 x}{9} - \frac{7}{3} + \frac{5 \cdot 3}{3}$

Step 5.2.4

Combine the numerators over the common denominator.

$y = - \frac{7 x}{9} + \frac{- 7 + 5 \cdot 3}{3}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $5$ by $3$ .

$y = - \frac{7 x}{9} + \frac{- 7 + 15}{3}$

Step 5.2.5.2

Add $- 7$ and $15$ .

$y = - \frac{7 x}{9} + \frac{8}{3}$

Step 6

The final answer is the equation in slope-intercept form.

$y = - \frac{7}{9} x + \frac{8}{3}$

Step 7

Replace $y$ by $f (x)$ .

$f (x) = - \frac{7}{9} x + \frac{8}{3}$

Step 8