Find the slope, use a point, then tidy into whichever form the question wants.
The problem
Find the equation of the line through (2, −3) and (6, 5).
m = (5 − (−3)) ÷ (6 − 2) = 84 = 2. Then y + 3 = 2(x − 2) gives y = 2x − 7. A flips the slope — it is rise over run. B reads the y-intercept off the point, but (2, −3) is not on the y-axis.
m = y₂ − y₁x₂ − x₁
m = 5 − (−3)6 − 2
watch the double negative
= 84
5 + 3 on top
m = 2
up 2 for every 1 across
Whichever point you call first, use it first on both top and bottom. Mixing them flips the sign and the line leans the wrong way.
y − y₁ = m(x − x₁)
y − (−3) = 2(x − 2)
using (2, −3)
y + 3 = 2x − 4
tidy both sides
y = 2x − 7
− 3 both sides
Using (6, 5) instead gives y − 5 = 2(x − 6), which tidies to the same line.
| Form | Looks like | Use it when |
|---|---|---|
| slope-intercept | y = mx + b | graphing, or reading off the slope |
| point-slope | y − y₁ = m(x − x₁) | you have a point and a slope |
| standard | Ax + By = C | the answer choices are written that way |
| intercept | xa + yb = 1 | both intercepts are given |
y = 2x − 7
slope-intercept
2x − y = 7
the same line, standard form
They are all the same line. If your answer is not among the choices, try rearranging before you assume it is wrong.
rises steeply, left to right
crosses the y-axis at −7
For the x-intercept, set y = 0: 2x = 7, so x = 3.5.
same slope
m = 2
negative reciprocal
m = −½
Flip it and change the sign. The two slopes always multiply to −1.
| Line | Slope | Perpendicular slope |
|---|---|---|
| y = 3x + 1 | 3 | −⅓ |
| y = −¾x | −¾ | ⁴⁄₃ |
| horizontal, y = 4 | 0 | undefined — vertical |
A fraction slope. The line through (−1, 2) and (3, 5), in standard form.
m = (5 − 2) ÷ (3 − (−1)) = 34
not a whole number this time
y − 2 = 34(x + 1)
point-slope with (−1, 2)
4y − 8 = 3x + 3
× 4 clears the fraction
3x − 4y = −11
check (3, 5): 9 − 20 = −11 ✓
What is different: multiply by the denominator of the slope before tidying, so standard form ends with whole numbers.
Same y twice. The line through (−4, 6) and (5, 6).
m = (6 − 6) ÷ (5 − (−4)) = 0 ÷ 9 = 0
a 0 on top is allowed
y − 6 = 0(x + 4)
point-slope still works
y = 6
no x term at all
What is different: a horizontal line has slope 0 and its equation is just y = the shared y-value.
Same x twice. The line through (2, −1) and (2, 7).
m = (7 − (−1)) ÷ (2 − 2) = 8 ÷ 0
a 0 on the bottom is not allowed
every point on it has x = 2
it runs straight up
x = 2
no slope, no y-intercept
What is different: a vertical line cannot be written as y = mx + b at all; its equation is x = the shared x-value.
The line through (2, −1) and (2, 7) is not y = 2 — that is a horizontal line. Whichever coordinate the two points share names the equation.
Worked: the line has x-intercept 5 and y-intercept −2. Write it in standard form and give its slope.
x5 + y−2 = 1
intercept form: x over a, y over b
2x − 5y = 10
× 10 clears both fractions
slope = −AB = −2−5 = 25
from Ax + By = C, no rearranging needed
Check the intercepts: y = 0 gives 2x = 10, x = 5; x = 0 gives −5y = 10, y = −2 ✓.
A letter in the equation: the line 2x + ky = 8 passes through (1, 3). Find its slope.
2(1) + k(3) = 8, so 3k = 6, k = 2
a point on the line makes the equation true
2x + 2y = 8 → slope = −2 ÷ 2 = −1
find the letter first, then read −A ÷ B
Backwards: a line of slope −25 cuts off a triangle of area 20 with the positive axes. Find the line.
intercepts (a, 0) and (0, b): slope = −ba
down b while going across a
ba = 25, so b = 2a5
from the slope
½ · a · b = 20, so a · 2a5 = 40
the legs of the triangle are the intercepts
a² = 100, a = 10, b = 4
positive axes, so take the positive root
x10 + y4 = 1 → 2x + 5y = 20
× 20; check area ½ · 10 · 4 = 20 ✓
The triangle's area is half of intercept × intercept — dropping the ½ doubles the answer. And xa + yb = 1 puts the x-intercept under x: writing 4x + 10y = … would swap the two intercepts.
Line questions want the answer in standard form and often set it up through an intercept, a parallel line or a triangle with the axes. Build the line with point-slope, clear fractions, then spend ten seconds substituting both given points (or both intercepts) into your choice.
Testing only the point you substituted proves nothing — check the other one.
Standard form
Find the equation of the line through (−1, 4) and (3, −4), in standard form.
m = (−4 − 4) ÷ (3 + 1) = −2, so y − 4 = −2(x + 1) gives y = −2x + 2. Move the x term across: 2x + y = 2. A loses a sign in the rearranging — check by putting (−1, 4) back in: 2(−1) + 4 = 2. ✓
