Find the Parallel Line (5,-2) that is parallel to the line 5x+6y=7

$(5, - 2)$ that is parallel to the line $5 x + 6 y = 7$

Step 1

Rewrite in slope-intercept form.

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

$y = m x + b$

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

$6 y = 7 - 5 x$

Divide each term in $6 y = 7 - 5 x$ by $6$ and simplify.

Divide each term in $6 y = 7 - 5 x$ by $6$ .

$\frac{6 y}{6} = \frac{7}{6} + \frac{- 5 x}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 y}{6} = \frac{7}{6} + \frac{- 5 x}{6}$

Divide $y$ by $1$ .

$y = \frac{7}{6} + \frac{- 5 x}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{7}{6} - \frac{5 x}{6}$

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

Reorder $\frac{7}{6}$ and $- \frac{5 x}{6}$ .

$y = - \frac{5 x}{6} + \frac{7}{6}$

Reorder terms.

$y = - (\frac{5}{6} x) + \frac{7}{6}$

Remove parentheses.

$y = - \frac{5}{6} x + \frac{7}{6}$

Step 2

Using the slope-intercept form, the slope is $- \frac{5}{6}$ .

$m = - \frac{5}{6}$

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 $- \frac{5}{6}$ and a given point $(5, - 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 5

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

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

Step 6

Solve for $y$ .

Simplify $- \frac{5}{6} \cdot (x - 5)$ .

Rewrite.

$y + 2 = 0 + 0 - \frac{5}{6} \cdot (x - 5)$

Simplify by adding zeros.

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

Apply the distributive property.

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

Combine $x$ and $\frac{5}{6}$ .

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

Multiply $- \frac{5}{6} \cdot - 5$ .

Multiply $- 5$ by $- 1$ .

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

Combine $5$ and $\frac{5}{6}$ .

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

Multiply $5$ by $5$ .

$y + 2 = - \frac{x \cdot 5}{6} + \frac{25}{6}$

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

$y + 2 = - \frac{5 x}{6} + \frac{25}{6}$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{5 x}{6} + \frac{25}{6} - 2$

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

$y = - \frac{5 x}{6} + \frac{25}{6} - 2 \cdot \frac{6}{6}$

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

$y = - \frac{5 x}{6} + \frac{25}{6} + \frac{- 2 \cdot 6}{6}$

Combine the numerators over the common denominator.

$y = - \frac{5 x}{6} + \frac{25 - 2 \cdot 6}{6}$

Simplify the numerator.

Multiply $- 2$ by $6$ .

$y = - \frac{5 x}{6} + \frac{25 - 12}{6}$

Subtract $12$ from $25$ .

$y = - \frac{5 x}{6} + \frac{13}{6}$

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

Reorder terms.

$y = - (\frac{5}{6} x) + \frac{13}{6}$

Remove parentheses.

$y = - \frac{5}{6} x + \frac{13}{6}$

Step 7

Find the Parallel Line (5,-2) that is parallel to the line 5x+6y=7

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