Find the Parallel Line x+3y=5 , (-2,5)

$x + 3 y = 5$ , $(- 2, 5)$

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 $x$ from both sides of the equation.

$3 y = 5 - x$

Divide each term in $3 y = 5 - x$ by $3$ and simplify.

Divide each term in $3 y = 5 - x$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

Reorder $\frac{5}{3}$ and $- \frac{x}{3}$ .

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

Reorder terms.

$y = - (\frac{1}{3} x) + \frac{5}{3}$

Remove parentheses.

$y = - \frac{1}{3} x + \frac{5}{3}$

Step 2

Using the slope-intercept form, the slope is $- \frac{1}{3}$ .

$m = - \frac{1}{3}$

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

$y - (5) = - \frac{1}{3} \cdot (x - (- 2))$

Step 5

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

$y - 5 = - \frac{1}{3} \cdot (x + 2)$

Step 6

Solve for $y$ .

Simplify $- \frac{1}{3} \cdot (x + 2)$ .

Rewrite.

$y - 5 = 0 + 0 - \frac{1}{3} \cdot (x + 2)$

Simplify by adding zeros.

$y - 5 = - \frac{1}{3} \cdot (x + 2)$

Apply the distributive property.

$y - 5 = - \frac{1}{3} x - \frac{1}{3} \cdot 2$

Combine $x$ and $\frac{1}{3}$ .

$y - 5 = - \frac{x}{3} - \frac{1}{3} \cdot 2$

Multiply $- \frac{1}{3} \cdot 2$ .

Multiply $2$ by $- 1$ .

$y - 5 = - \frac{x}{3} - 2 (\frac{1}{3})$

Combine $- 2$ and $\frac{1}{3}$ .

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

Move the negative in front of the fraction.

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

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

Add $5$ to both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $5$ by $3$ .

$y = - \frac{x}{3} + \frac{- 2 + 15}{3}$

Add $- 2$ and $15$ .

$y = - \frac{x}{3} + \frac{13}{3}$

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

Reorder terms.

$y = - (\frac{1}{3} x) + \frac{13}{3}$

Remove parentheses.

$y = - \frac{1}{3} x + \frac{13}{3}$

Step 7

Find the Parallel Line x+3y=5 , (-2,5)

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