AI Tutor
Ever look down a straight stretch of train tracks? The two steel rails travel for miles side-by-side, perfectly locked in direction so they never cross.
In geometry, parallel lines are coplanar lines that never intersect because they share the exact same steepness, or slope (m1=m2m_1 = m_2).
If two parallel lines have the same steepness, how do we write the exact equation for a new line when we only know the old line and one target point?

The Parallel Blueprint

Every line equation needs two pieces of information: a slope (mm) and a point (x1,y1)(x_1, y_1) on the line.
We use point-slope form, y−y1=m(x−x1)y - y_1 = m(x - x_1), which comes directly from the slope definition m=y−y1x−x1m = \frac{y - y_1}{x - x_1} by multiplying both sides by (x−x1)(x - x_1).
Let's test this recipe on a standard line equation where the slope is already visible.

Worked Example: Given Slope-Intercept Form

Find the equation of the line parallel to y=3x−5y = 3x - 5 that passes through the point (2,4)(2, 4).
First, identify the slope of the given line: m=3m = 3. Because parallel lines share identical slopes, our new line also has m=3m = 3.
Next, substitute m=3m = 3 and (x1,y1)=(2,4)(x_1, y_1) = (2, 4) into point-slope form: