Find the Parallel Line What is an equation of the line that passes through the point (1,3) and is parallel to the line 3x+y=9 ?

What is an equation of the line that passes through the point $(1, 3)$ and is parallel to the line $3 x + y = 9$ ?

Step 1

Write the problem as a mathematical expression.

$(1, 3)$ , $3 x + y = 9$

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 $3 x$ from both sides of the equation.

$y = 9 - 3 x$

Reorder $9$ and $- 3 x$ .

$y = - 3 x + 9$

Step 3

Using the slope-intercept form, the slope is $- 3$ .

$m = - 3$

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

$y - (3) = - 3 \cdot (x - (1))$

Step 6

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

$y - 3 = - 3 \cdot (x - 1)$

Step 7

Solve for $y$ .

Simplify $- 3 \cdot (x - 1)$ .

Rewrite.

$y - 3 = 0 + 0 - 3 \cdot (x - 1)$

Simplify by adding zeros.

$y - 3 = - 3 \cdot (x - 1)$

Apply the distributive property.

$y - 3 = - 3 x - 3 \cdot - 1$

Multiply $- 3$ by $- 1$ .

$y - 3 = - 3 x + 3$

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

Add $3$ to both sides of the equation.

$y = - 3 x + 3 + 3$

Add $3$ and $3$ .

$y = - 3 x + 6$

Step 8

Find the Parallel Line What is an equation of the line that passes through the point (1,3) and is parallel to the line 3x+y=9 ?

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