Precalculus Examples

$5 x + 4 y = 20$ $(3, 2)$

Step 1

Rewrite in slope-intercept form.

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Step 1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 1.2

Subtract $5 x$ from both sides of the equation.

$4 y = 20 - 5 x$

Step 1.3

Divide each term in $4 y = 20 - 5 x$ by $4$ and simplify.

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Step 1.3.1

Divide each term in $4 y = 20 - 5 x$ by $4$ .

$\frac{4 y}{4} = \frac{20}{4} + \frac{- 5 x}{4}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{20}{4} + \frac{- 5 x}{4}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{20}{4} + \frac{- 5 x}{4}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Divide $20$ by $4$ .

$y = 5 + \frac{- 5 x}{4}$

Step 1.3.3.1.2

Move the negative in front of the fraction.

$y = 5 - \frac{5 x}{4}$

Step 1.4

Write in $y = m x + b$ form.

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Step 1.4.1

Reorder $5$ and $- \frac{5 x}{4}$ .

$y = - \frac{5 x}{4} + 5$

Step 1.4.2

Reorder terms.

$y = - (\frac{5}{4} x) + 5$

Step 1.4.3

Remove parentheses.

$y = - \frac{5}{4} x + 5$

Step 2

Using the slope-intercept form, the slope is $- \frac{5}{4}$ .

$m = - \frac{5}{4}$

Step 3

To find an equation that is parallel, the slopes must be equal. Find the parallel line using the point-slope formula.

Step 4

Use the slope $- \frac{5}{4}$ and a given point $(3, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{5}{4} \cdot (x - (3))$

Step 5

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

$y - 2 = - \frac{5}{4} \cdot (x - 3)$

Step 6

Solve for $y$ .

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Step 6.1

Simplify $- \frac{5}{4} \cdot (x - 3)$ .

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Step 6.1.1

Rewrite.

$y - 2 = 0 + 0 - \frac{5}{4} \cdot (x - 3)$

Step 6.1.2

Simplify by adding zeros.

$y - 2 = - \frac{5}{4} \cdot (x - 3)$

Step 6.1.3

Apply the distributive property.

$y - 2 = - \frac{5}{4} x - \frac{5}{4} \cdot - 3$

Step 6.1.4

Combine $x$ and $\frac{5}{4}$ .

$y - 2 = - \frac{x \cdot 5}{4} - \frac{5}{4} \cdot - 3$

Step 6.1.5

Multiply $- \frac{5}{4} \cdot - 3$ .

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Step 6.1.5.1

Multiply $- 3$ by $- 1$ .

$y - 2 = - \frac{x \cdot 5}{4} + 3 (\frac{5}{4})$

Step 6.1.5.2

Combine $3$ and $\frac{5}{4}$ .

$y - 2 = - \frac{x \cdot 5}{4} + \frac{3 \cdot 5}{4}$

Step 6.1.5.3

Multiply $3$ by $5$ .

$y - 2 = - \frac{x \cdot 5}{4} + \frac{15}{4}$

Step 6.1.6

Move $5$ to the left of $x$ .

$y - 2 = - \frac{5 x}{4} + \frac{15}{4}$

Step 6.2

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

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Step 6.2.1

Add $2$ to both sides of the equation.

$y = - \frac{5 x}{4} + \frac{15}{4} + 2$

Step 6.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{5 x}{4} + \frac{15}{4} + 2 \cdot \frac{4}{4}$

Step 6.2.3

Combine $2$ and $\frac{4}{4}$ .

$y = - \frac{5 x}{4} + \frac{15}{4} + \frac{2 \cdot 4}{4}$

Step 6.2.4

Combine the numerators over the common denominator.

$y = - \frac{5 x}{4} + \frac{15 + 2 \cdot 4}{4}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $2$ by $4$ .

$y = - \frac{5 x}{4} + \frac{15 + 8}{4}$

Step 6.2.5.2

Add $15$ and $8$ .

$y = - \frac{5 x}{4} + \frac{23}{4}$

Step 6.3

Write in $y = m x + b$ form.

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Step 6.3.1

Reorder terms.

$y = - (\frac{5}{4} x) + \frac{23}{4}$

Step 6.3.2

Remove parentheses.

$y = - \frac{5}{4} x + \frac{23}{4}$

Step 7