Coordinate Geometry • Topic 3 of 5

Equations of Lines

The most useful form on the GRE is slope-intercept form, y = mx + b, because you can read the slope m and the y-intercept b directly. To build a line's equation you need a slope and one point: find m, then substitute the point's coordinates to solve for b. To find where two lines cross, set their equations equal (or solve the system) — the intersection is the point satisfying both. Intercepts are quick: set y = 0 to find the x-intercept, set x = 0 to find the y-intercept. When an equation arrives in the form Ax + By = C, rearranging to y = mx + b makes the slope and intercept visible. Watch for the special cases: y = k is a horizontal line (slope 0) and x = k is a vertical line (undefined slope) — these are not in y = mx + b form, and mislabelling their slopes is a frequent GRE trap.

The line y equals two x plus one drawn on axes with its y-intercept at (0, 1) markedy = mx + bxy-4-3-2-11234-5-4-3-2-112345O(0, 1)y = 2x + 1b is the y-interceptslope m = 2, y-intercept b = 1

✅ Solved examples

1. Write the equation of the line with slope 2 passing through (3, 5).
y = 2x + b. Substitute (3, 5): 5 = 2(3) + b, so b = −1. The line is y = 2x − 1.
2. Find the x-intercept of y = 3x − 12.
Set y = 0: 0 = 3x − 12, so 3x = 12 and x = 4. The x-intercept is (4, 0).
3. Put 2x + 4y = 8 into slope-intercept form and state the slope.
4y = −2x + 8, so y = −(1/2)x + 2. The slope is −1/2 and the y-intercept is 2.
4. Where do y = x + 1 and y = −x + 5 intersect?
Set equal: x + 1 = −x + 5, so 2x = 4, x = 2. Then y = 2 + 1 = 3. They meet at (2, 3).

✏️ Practice — try these, take hints as needed

1. Write the line with slope 3 through (1, 4).
y = 3x + b.
Substitute (1, 4): 4 = 3 + b.
Solve for b.
y = 3x + 1
2. Find the y-intercept of y = −2x + 7.
b is the y-intercept.
Read it off the equation.
x = 0 gives y = 7.
7 (point (0, 7))
3. Find the x-intercept of y = 4x − 8.
Set y = 0.
0 = 4x − 8.
4x = 8.
x = 2 (point (2, 0))
4. Convert 3x − y = 6 to slope-intercept form.
Solve for y.
−y = −3x + 6.
Multiply through by −1.
y = 3x − 6
5. Where do y = 2x and y = x + 3 intersect?
Set 2x = x + 3.
x = 3.
Find y from either equation.
(3, 6)

📝 Topic test — 8 questions

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