Find Any Equation Parallel to the Line 2x+3y=7

$2 x + 3 y = 7$

Step 1

Choose a point that the parallel line will pass through.

$(0, 0)$

Step 2

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 $2 x$ from both sides of the equation.

$3 y = 7 - 2 x$

Divide each term in $3 y = 7 - 2 x$ by $3$ and simplify.

Divide each term in $3 y = 7 - 2 x$ by $3$ .

$\frac{3 y}{3} = \frac{7}{3} + \frac{- 2 x}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{7}{3} + \frac{- 2 x}{3}$

Divide $y$ by $1$ .

$y = \frac{7}{3} + \frac{- 2 x}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$y = \frac{7}{3} - \frac{2 x}{3}$

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

Reorder $\frac{7}{3}$ and $- \frac{2 x}{3}$ .

$y = - \frac{2 x}{3} + \frac{7}{3}$

Reorder terms.

$y = - (\frac{2}{3} x) + \frac{7}{3}$

Remove parentheses.

$y = - \frac{2}{3} x + \frac{7}{3}$

Step 3

Using the slope-intercept form, the slope is $- \frac{2}{3}$ .

$m = - \frac{2}{3}$

Step 4

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

Step 5

Use the slope $- \frac{2}{3}$ and a given point $(0, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 6

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

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

Step 7

Solve for $y$ .

Add $y$ and $0$ .

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

Simplify $- \frac{2}{3} \cdot (x + 0)$ .

Add $x$ and $0$ .

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

Combine $x$ and $\frac{2}{3}$ .

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

Move $2$ to the left of $x$ .

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

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

Reorder terms.

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

Remove parentheses.

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

Step 8

Find Any Equation Parallel to the Line 2x+3y=7

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