Skill: Write the equation of a parallel line

Parallel lines have the same slope but usually different $y$-intercepts. To write the equation of a parallel line:

1. Find the slope of the given line.
2. Use that same slope for the new line.
3. Substitute the given point into point-slope form:

$$y-y_1=m(x-x_1)$$

4. Simplify if needed.

Example

Write the equation of the line parallel to

$$y=2x-5$$

that passes through $(3,4)$.

Step 1: Find the slope.

The equation is in slope-intercept form, $y=mx+b$, so the slope is

$$m=2.$$

Step 2: Use the same slope.

A parallel line must also have slope $2$.

Step 3: Use point-slope form with $(x_1,y_1)=(3,4)$.

$$y-4=2(x-3)$$

Step 4: Simplify.

$$y-4=2x-6$$ $$y=2x-2$$

Therefore, the equation of the parallel line is

$$\boxed{y=2x-2}.$$

Both lines have slope $2$, so they are parallel. Their intercepts are different: $-5$ and $-2$.

Remember: perpendicular lines do not have the same slope; their slopes are negative reciprocals. Horizontal lines are parallel when they have the same equation form $y=c$, while vertical lines are parallel when they have the same form $x=c$.