Math

Slope and Slope-Intercept Form

2 min readEasy5-question drill

Slope-intercept form is the single most tested idea in SAT math — it shows up in graphs, word problems, function questions, and systems. Master this one format and a huge chunk of the test opens up.

Where this lesson fits

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→Linear Functions →Linear Equations in 2 Variables →Systems of 2 Linear Equations in 2 Variables

A linear equation is just an equation whose graph is a straight line. The cleanest way to write one is slope-intercept form:

y = mx + b

This little formula has two stars:

m is the slope — how steep the line is, and which way it tilts. Slope tells you how much y changes every time x goes up by 1.
b is the y-intercept — the y-value where the line crosses the vertical y-axis (this happens when x = 0).

What slope really means. Slope is rise over run:

m = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁)

If you have two points, subtract the y's on top, subtract the x's on the bottom (in the same order), and divide. A positive slope goes uphill left-to-right; a negative slope goes downhill. A slope of 0 is a flat horizontal line.

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A line with slope 1.5 and y-intercept 2. The rise/run staircase shows how slope is measured.

Reading a line fast. Given y = 3x + 5, you instantly know: slope is 3 (line rises 3 for every 1 right), and it crosses the y-axis at (0, 5). No graphing needed.

Function notation is the same thing in disguise. The test loves to write f(x) = 4x − 3. That f(x) is just a fancy name for y. So f(x) = 4x − 3 is a line with slope 4 and y-intercept −3. When a question asks for f(4), it means: plug in x = 4.

f(4) = 4(4) − 3 = 16 − 3 = 13

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The function f(x) = 4x − 3: slope 4, y-intercept −3.

Building the equation. If a word problem says "a plumber charges $40 to show up plus $25 per hour," the $40 is the starting amount (b, the y-intercept) and $25 is the rate of change (m, the slope). So y = 25x + 40, where x is hours.

The key habit: whenever you see a straight-line relationship, hunt for the starting value (b) and the rate of change (m). Almost every linear question is testing whether you can find those two numbers.

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Quick check

If f(x) = 4x + (-3), what is f(4)?

Worked examples

Example 1

If f(x) = 5x + (-3), what is f(7)?

Example 2

A line passes through the points (2, 1) and (6, 13). What is the slope of the line?

Example 3

A water tank starts with 30 gallons and is filled at a constant rate of 8 gallons per minute. Which equation gives the number of gallons y in the tank after x minutes?

Common pitfalls

Swapping slope and y-intercept in word problems

The per-unit rate is always the slope (m) and the one-time starting amount is the y-intercept (b). A flat fee plus an hourly rate becomes y = (rate)x + (fee), not the other way around.

Subtracting points in different orders

When using m = (y₂ − y₁)/(x₂ − x₁), you must subtract the y's and x's in the SAME order. Flipping one but not the other gives the wrong sign on your slope.

Forgetting to divide for slope

Students compute the rise (top) and stop, choosing that number as the slope. Slope is rise DIVIDED BY run — always finish the division.

Misreading f(x) notation

f(4) does not mean multiply f by 4. It means substitute 4 for x. Treat f(x) exactly like y.

Key takeaways

Slope-intercept form is y = mx + b: m is the slope, b is the y-intercept (where the line crosses the y-axis at x = 0).
Slope = rise/run = (y₂ − y₁)/(x₂ − x₁) — subtract in the same order on top and bottom, then divide.
In a word problem, the constant 'per' rate is the slope and the initial/starting amount is the y-intercept.
f(x) is just another name for y; f(number) means plug that number in for x.
Positive slope rises left-to-right, negative slope falls, zero slope is horizontal.

Try it yourself

5 practice questions on Slope and Slope-Intercept Form, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Slope and Slope-Intercept Form

You are here

→Linear Functions →Linear Equations in 2 Variables →Systems of 2 Linear Equations in 2 Variables

Worked examples

Common pitfalls

Key takeaways

Further reading

Try it yourself