Skill: Determine equations of parallel lines

To find an equation of a line parallel to a given line:

1. Identify the slope of the given line.
2. Use the same slope for the new line.
3. Substitute the given point into the equation to find the intercept, or use point-slope form.
4. Check that the new line is distinct from the original line.

Example: Find the equation of the line parallel to

$$y=-2x+5$$

that passes through $(3,4)$.

The given line is already in slope-intercept form, $y=mx+b$, so its slope is

$$m=-2.$$

A parallel line must have the same slope. Using point-slope form,

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

with $(x_1,y_1)=(3,4)$ and $m=-2$:

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

Simplify:

$$y-4=-2x+6$$ $$y=-2x+10.$$

Therefore, the equation is

$$\boxed{y=-2x+10}.$$

Both lines have slope $-2$, so they are parallel. They are not the same line because their $y$-intercepts, $5$ and $10$, are different.

For special cases, horizontal parallel lines have the same equation form $y=c$ and slope $0$. Vertical parallel lines have the form $x=c$; for example, $x=2$ and $x=7$ are parallel because both are vertical and have different $x$-values.