Rise over run.
m = y₂ − y₁x₂ − x₁
A horizontal line has slope 0 — no rise. A vertical line has undefined slope, because the run is 0 and you cannot divide by zero. Those two are opposite, not the same.
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- Slope and Forms of Linear Equations
Slope and Forms of Linear Equations
Slope is rise over run — and in a real situation, it is the rate.
Find the slope
What is the slope of the line through (−2, 7) and (4, −5)?
Slope = y₂ − y₁x₂ − x₁ = −5 − 74 − (−2) = −126 = −2. The line falls as it goes right, so a negative slope is expected. C flips the fraction, putting the run over the rise. B loses the sign — but the y-value dropped from 7 to −5, so the slope must be negative.
Order does not matter — as long as you are consistent.
−5 − 74 − (−2)
= −126 = −2
✓
7 − (−5)−2 − 4
= 12−6 = −2
same ✓
7 − (−5)4 − (−2)
= 126 = 2
points swapped on one line only ✗
Start with the same point on both top and bottom. Mixing them is what turns a negative slope into a positive one.
Slope-intercept form shows the line at a glance.
y = mx + b — m is the slope, b is where it crosses the y-axis.
slope −2, through (4, −5)
find b
−5 = −2(4) + b
substitute the point
−5 = −8 + b, so b = 3
solve
y = −2x + 3
the equation
Check with the other point: −2(−2) + 3 = 7 ✓ — that is (−2, 7), so both points fit.
Point-slope form saves the substitution step.
y − y₁ = m(x − x₁)
m = −2, point (4, −5)
drop them straight in
y + 5 = −2(x − 4)
y − (−5) is y + 5
y + 5 = −2x + 8
expand
y = −2x + 3
same answer as step 3
| Form | Looks like | Best for |
|---|---|---|
| Slope-intercept | y = mx + b | graphing, reading the slope |
| Point-slope | y − y₁ = m(x − x₁) | building from a point and a slope |
| Standard | Ax + By = C | finding intercepts |
In a real situation, slope is a rate.
A plumber charges ₱500 to come out
plus ₱350 an hour
C = 350h + 500
slope 350, intercept 500
slope = cost per extra hour
the rate of change
intercept = cost of zero hours
the call-out fee
Being able to say what m and b mean is more useful than computing them. The slope is always "per one unit of x", and the intercept is always the starting value.
Slope from standard form, and parallel or perpendicular lines.
Parallel lines share a slope. Perpendicular slopes are negative reciprocals: flip AND change the sign.
Find the slope of 3x − 5y = 15.
−5y = −3x + 15
move the x term
y = (35)x − 3
÷ (−5), every term
slope = 35
the coefficient of x
The x-coefficient in standard form is not the slope
Reading 3x − 5y = 15 as "slope 3" skips the rearranging. The slope only shows once y stands alone. Shortcut for Ax + By = C: m = −AB = −3−5 = 35.
Find the line through (2, −1) parallel to y = −4x + 6.
parallel → m = −4
same slope
y + 1 = −4(x − 2)
point-slope
y = −4x + 7
expand, − 1
Check: −4(2) + 7 = −1 ✓. The 6 of the old line is irrelevant; only the slope carries over.
Find the slope of a line perpendicular to y = (34)x − 2.
34 → flip → 43
reciprocal
change the sign → −43
negative reciprocal
Test: (34) × (−43) = −1. Perpendicular slopes always multiply to −1. Writing 43 (flip only) or −34 (sign only) is the usual half-done answer.
Variations: a vertical line, a fraction slope and intercept, back to standard form.
The same formulas, at their edges.
The line through (3, −2) and (3, 6)
m = 6 − (−2)3 − 3 = 80
the run is zero
undefined slope; equation x = 3
a vertical line
A vertical line has no y = mx + b form at all. Its equation names the x every point shares.
The line through (−1, 4) and (5, 0)
m = 0 − 45 − (−1) = −46 = −23
reduce
0 = (−23)(5) + b
substitute (5, 0)
b = 103
0 = −103 + b
y = (−23)x + 103
Check (−1, 4): 23 + 103 = 123 = 4 ✓. Fraction intercepts are normal; keep them exact.
Write y = (23)x − 4 in standard form.
3y = 2x − 12
× 3 to clear the fraction
2x − 3y = 12
x and y on the left, A positive
Check with (6, 0): 2(6) − 0 = 12 ✓, and (23)(6) − 4 = 0 ✓.
On the exam
Items ask for a slope from two points or from an equation, an equation from a slope and a point, the slope of a parallel or perpendicular line, or what the slope means in a peso situation. The fastest route is to get y on its own first, so m and b can be read off. For a 10-second check, put one known point into your final equation.
Now you try
Practice problem
Write the equation
A line has slope 3 and passes through (2, 1). What is its equation in slope-intercept form?
Substitute: 1 = 3(2) + b, so 1 = 6 + b and b = −5, giving y = 3x − 5. Check: 3(2) − 5 = 1 ✓ A treats the point's y-value as the intercept, which would only be right if the point were on the y-axis. C has the sign of b wrong.
Hint: b is the y-value when x = 0, and this point has x = 2. Put the point into y = 3x + b and solve for b.