Precalculus Examples

$6 - \frac{8 y + 5 x}{2} = 4$ , $(3, 2)$

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

Simplify the left side.

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Step 1.2.1

Simplify $6 - \frac{8 y + 5 x}{2}$ .

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Step 1.2.1.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 \cdot \frac{2}{2} - \frac{8 y + 5 x}{2} = 4$

Step 1.2.1.2

Simplify terms.

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Step 1.2.1.2.1

Combine $6$ and $\frac{2}{2}$ .

$\frac{6 \cdot 2}{2} - \frac{8 y + 5 x}{2} = 4$

Step 1.2.1.2.2

Combine the numerators over the common denominator.

$\frac{6 \cdot 2 - (8 y + 5 x)}{2} = 4$

Step 1.2.1.3

Simplify the numerator.

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Step 1.2.1.3.1

Multiply $6$ by $2$ .

$\frac{12 - (8 y + 5 x)}{2} = 4$

Step 1.2.1.3.2

Apply the distributive property.

$\frac{12 - (8 y) - (5 x)}{2} = 4$

Step 1.2.1.3.3

Multiply $8$ by $- 1$ .

$\frac{12 - 8 y - (5 x)}{2} = 4$

Step 1.2.1.3.4

Multiply $5$ by $- 1$ .

$\frac{12 - 8 y - 5 x}{2} = 4$

Step 1.3

Multiply both sides by $2$ .

$\frac{12 - 8 y - 5 x}{2} \cdot 2 = 4 \cdot 2$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Simplify $\frac{12 - 8 y - 5 x}{2} \cdot 2$ .

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Step 1.4.1.1.1

Cancel the common factor of $2$ .

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Step 1.4.1.1.1.1

Cancel the common factor.

$\frac{12 - 8 y - 5 x}{2} \cdot 2 = 4 \cdot 2$

Step 1.4.1.1.1.2

Rewrite the expression.

$12 - 8 y - 5 x = 4 \cdot 2$

Step 1.4.1.1.2

Simplify the expression.

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Step 1.4.1.1.2.1

Move $12$ .

$- 8 y - 5 x + 12 = 4 \cdot 2$

Step 1.4.1.1.2.2

Reorder $- 8 y$ and $- 5 x$ .

$- 5 x - 8 y + 12 = 4 \cdot 2$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Multiply $4$ by $2$ .

$- 5 x - 8 y + 12 = 8$

Step 1.5

Solve for $y$ .

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Step 1.5.1

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

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Step 1.5.1.1

Add $5 x$ to both sides of the equation.

$- 8 y + 12 = 8 + 5 x$

Step 1.5.1.2

Subtract $12$ from both sides of the equation.

$- 8 y = 8 + 5 x - 12$

Step 1.5.1.3

Subtract $12$ from $8$ .

$- 8 y = 5 x - 4$

Step 1.5.2

Divide each term in $- 8 y = 5 x - 4$ by $- 8$ and simplify.

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Step 1.5.2.1

Divide each term in $- 8 y = 5 x - 4$ by $- 8$ .

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Simplify each term.

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Step 1.5.2.3.1.1

Move the negative in front of the fraction.

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

Step 1.5.2.3.1.2

Cancel the common factor of $- 4$ and $- 8$ .

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Step 1.5.2.3.1.2.1

Factor $- 4$ out of $- 4$ .

$y = - \frac{5 x}{8} + \frac{- 4 (1)}{- 8}$

Step 1.5.2.3.1.2.2

Cancel the common factors.

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Step 1.5.2.3.1.2.2.1

Factor $- 4$ out of $- 8$ .

$y = - \frac{5 x}{8} + \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 1.5.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{5 x}{8} + \frac{- 4 \cdot 1}{- 4 \cdot 2}$

Step 1.5.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{5 x}{8} + \frac{1}{2}$

Step 1.6

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

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Step 1.6.1

Reorder terms.

$y = - (\frac{5}{8} x) + \frac{1}{2}$

Step 1.6.2

Remove parentheses.

$y = - \frac{5}{8} x + \frac{1}{2}$

Step 2

Using the slope-intercept form, the slope is $- \frac{5}{8}$ .

$m = - \frac{5}{8}$

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 $- \frac{5}{8}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = - \frac{5}{8} \cdot (x - (3))$

Step 5

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

$y - 2 = - \frac{5}{8} \cdot (x - 3)$

Step 6

Solve for $y$ .

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Step 6.1

Simplify $- \frac{5}{8} \cdot (x - 3)$ .

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Step 6.1.1

Rewrite.

$y - 2 = 0 + 0 - \frac{5}{8} \cdot (x - 3)$

Step 6.1.2

Simplify by adding zeros.

$y - 2 = - \frac{5}{8} \cdot (x - 3)$

Step 6.1.3

Apply the distributive property.

$y - 2 = - \frac{5}{8} x - \frac{5}{8} \cdot - 3$

Step 6.1.4

Combine $x$ and $\frac{5}{8}$ .

$y - 2 = - \frac{x \cdot 5}{8} - \frac{5}{8} \cdot - 3$

Step 6.1.5

Multiply $- \frac{5}{8} \cdot - 3$ .

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Step 6.1.5.1

Multiply $- 3$ by $- 1$ .

$y - 2 = - \frac{x \cdot 5}{8} + 3 (\frac{5}{8})$

Step 6.1.5.2

Combine $3$ and $\frac{5}{8}$ .

$y - 2 = - \frac{x \cdot 5}{8} + \frac{3 \cdot 5}{8}$

Step 6.1.5.3

Multiply $3$ by $5$ .

$y - 2 = - \frac{x \cdot 5}{8} + \frac{15}{8}$

Step 6.1.6

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

$y - 2 = - \frac{5 x}{8} + \frac{15}{8}$

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.

$y = - \frac{5 x}{8} + \frac{15}{8} + 2$

Step 6.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = - \frac{5 x}{8} + \frac{15}{8} + 2 \cdot \frac{8}{8}$

Step 6.2.3

Combine $2$ and $\frac{8}{8}$ .

$y = - \frac{5 x}{8} + \frac{15}{8} + \frac{2 \cdot 8}{8}$

Step 6.2.4

Combine the numerators over the common denominator.

$y = - \frac{5 x}{8} + \frac{15 + 2 \cdot 8}{8}$

Step 6.2.5

Simplify the numerator.

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Step 6.2.5.1

Multiply $2$ by $8$ .

$y = - \frac{5 x}{8} + \frac{15 + 16}{8}$

Step 6.2.5.2

Add $15$ and $16$ .

$y = - \frac{5 x}{8} + \frac{31}{8}$

Step 6.3

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

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Step 6.3.1

Reorder terms.

$y = - (\frac{5}{8} x) + \frac{31}{8}$

Step 6.3.2

Remove parentheses.

$y = - \frac{5}{8} x + \frac{31}{8}$

Step 7