Equation of a Line

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Skill: equation-of-a-line

Equation of a Line

An equation of a line is a compact description of every point on that line. Two numbers — the slope and the y-intercept — pin down infinitely many points, turning a picture into a prediction machine.

1 Three Forms, One Line

The most useful form is slope-intercept form. Read it like instructions: start at b on the y-axis, then climb m units for every 1 step right.

y = mx + b   m = slope (steepness + direction) · b = y-intercept (where it crosses the y-axis)
run = 1rise = 2b = 1: start herem = 2: climb from here(1, 3)
Read y = 2x + 1 like instructions: start at b = 1 on the y-axis, then climb m = 2 for every 1 step right.

The three costumes

Slope-intercept y = mx + b — easiest to graph: intercept first, slope triangle second.

Point-slope y − y1 = m(x − x1) — fastest to write when you know a slope and one point.

Standard Ax + By = C — tidiest for systems of equations (integers, A > 0).

They are the same line in different outfits. Point-slope is the stepping stone: write it first from data, then convert to whichever form the question asks for.

2 See It: Three Costumes, One Line

The points (−1, 2) and (3, 10) determine a single line. Here it is written three ways — the graph never changes.

(−1, 2)(3, 10)
One line, two points. The equation you write depends on what the question asks for.
Point-slope (write this first)
y − 2 = 2(x + 1)

Slope m = 2 from the two points; plug in (−1, 2). Watch the sign: x − (−1) = x + 1.

Slope-intercept (easiest to graph)
y = 2x + 4

Distribute and solve for y. Check both points: 2(3) + 4 = 10 ✓ and 2(−1) + 4 = 2 ✓.

Standard (tidy integers)
2x − y = −4

Move the x-term left, flip signs so A > 0. Same line — test (−1, 2): −2 − 2 = −4 ✓.

3 Worked Examples

The pattern: find the slope if you need it, write point-slope, then convert to the requested form. Always check with a point.

Example 1 Point-slope from slope and a point — slope 3 through (2, 7)
  1. Point-slope form is built for exactly this: y − y1 = m(x − x1).
  2. Plug in m = 3, (x1, y1) = (2, 7): y − 7 = 3(x − 2).
  3. Simplify to slope-intercept: y − 7 = 3x − 6 → y = 3x + 1.
y = 3x + 1. Check: does (2, 7) satisfy it? 3(2) + 1 = 7 ✓.
Example 2 From two points — through (−1, 2) and (3, 10)
  1. First find the slope: m = 10 − 23 − (−1) = 84 = 2.
  2. Use point-slope with either point — here (−1, 2): y − 2 = 2(x − (−1)). Parenthesize first!
  3. Simplify: y − 2 = 2(x + 1) → y − 2 = 2x + 2 → y = 2x + 4.
y = 2x + 4. Check BOTH points: 2(3) + 4 = 10 ✓ and 2(−1) + 4 = 2 ✓. Always check the point you didn’t use to build the equation.
Example 3 Directly — slope −1/2, y-intercept 5
  1. This is already m and b — just assemble y = mx + b.
  2. y = −12x + 5.
y = −(1/2)x + 5. Check: at x = 0, y = 5 (the intercept ✓); at x = 2, y = 4 — down 1 over 2 right, matching slope −1/2 ✓.
Example 4 Edge cases — horizontal and vertical
  1. A horizontal line through (2, 5) has slope 0: y = 0·x + 5, i.e. y = 5.
  2. A vertical line through (−3, 4) has undefined slope — no y = mx + b form exists; it is simply x = −3.
y = 5 contains (2, 5), (9, 5) — every point with y = 5 ✓. x = −3 contains (−3, 4), (−3, −100) ✓.

4 Common Mistakes

Three errors that show up on nearly every equations-of-lines quiz.

Mistake 1: Sign slip in point-slope with a negative coordinate
Wrong
Through (−1, 2) with slope 2: y − 2 = 2(x − 1) — the x1 sign got dropped.
Right
y − 2 = 2(x − (−1)) = 2(x + 1).
Fix: parenthesize first, simplify second. x − (x1) with x1 = −1 is x − (−1) = x + 1. Never do it mentally.
Mistake 2: Confusing slope and intercept
Wrong
“In y = 4x − 2, the slope is −2.”
Right
Slope m = 4 (the number attached to x); intercept b = −2 (the constant).
Memory hook: m multiplies x — it never stands alone. The lonely number is b.
Mistake 3: Stopping at point-slope and calling it done
Wrong
Leaving y − 5 = 2(x − 1) when the question asks for slope-intercept form.
Right
Simplify to y = 2x + 3.
Fix: read the question’s requested form. Point-slope is a stepping stone, not usually the final answer.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Write the equation (slope-intercept form) of the line with slope 2 through (1, 5).
Answer
Start with point-slope: y − 5 = 2(x − 1). Distribute: y − 5 = 2x − 2, so y = 2x + 3.
Check: 2(1) + 3 = 5 ✓.
2. Rewrite y = 2x + 3 in standard form (Ax + By = C, with A > 0).
Answer
Move the x-term left: −2x + y = 3. Multiply by −1 so A > 0: 2x − y = −3.
Check with (0, 3): 2(0) − 3 = −3 ✓.
3. Write the equation of the line with slope 4 and y-intercept −2.
Answer
Assemble y = mx + b with m = 4, b = −2: y = 4x − 2.
Check: at x = 0, y = −2 (the intercept ✓).

Key Points to Remember

  • y = mx + b: m is the slope (attached to x), b is the y-intercept (the constant).
  • y − y1 = m(x − x1): fastest from a slope and a point — parenthesize negative coordinates first.
  • Ax + By = C: standard form with integers and A > 0.
  • Horizontal line: y = k (slope 0). Vertical line: x = h (undefined slope, no y = mx + b form).
  • Always check with a point — preferably one you didn’t use to build the equation.
Practice line equations Try the form converter See it on FiboMap
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