Find the Parallel Line 3x+4y=5 (3,-2)

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

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 $3 x$ from both sides of the equation.

$4 y = 5 - 3 x$

Divide each term in $4 y = 5 - 3 x$ by $4$ and simplify.

Divide each term in $4 y = 5 - 3 x$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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

Reorder $\frac{5}{4}$ and $- \frac{3 x}{4}$ .

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

Reorder terms.

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

Remove parentheses.

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

Step 2

Using the slope-intercept form, the slope is $- \frac{3}{4}$ .

$m = - \frac{3}{4}$

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 $- \frac{3}{4}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 5

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

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

Step 6

Solve for $y$ .

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

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

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

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

Multiply $- 3$ by $- 1$ .

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

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

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

Multiply $3$ by $3$ .

$y + 2 = - \frac{x \cdot 3}{4} + \frac{9}{4}$

Move $3$ to the left of $x$ .

$y + 2 = - \frac{3 x}{4} + \frac{9}{4}$

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

Subtract $2$ from both sides of the equation.

$y = - \frac{3 x}{4} + \frac{9}{4} - 2$

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

$y = - \frac{3 x}{4} + \frac{9}{4} - 2 \cdot \frac{4}{4}$

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

$y = - \frac{3 x}{4} + \frac{9}{4} + \frac{- 2 \cdot 4}{4}$

Combine the numerators over the common denominator.

$y = - \frac{3 x}{4} + \frac{9 - 2 \cdot 4}{4}$

Simplify the numerator.

Multiply $- 2$ by $4$ .

$y = - \frac{3 x}{4} + \frac{9 - 8}{4}$

Subtract $8$ from $9$ .

$y = - \frac{3 x}{4} + \frac{1}{4}$

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

Reorder terms.

$y = - (\frac{3}{4} x) + \frac{1}{4}$

Remove parentheses.

$y = - \frac{3}{4} x + \frac{1}{4}$

Step 7

Find the Parallel Line 3x+4y=5 (3,-2)

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