Math

Linear Equations in 2 Variables

2 min readEasy5-question drill

Linear equations in two variables are everywhere on the math section — from word problems about taxi fares to graph questions about slope and intercepts. Master this one topic and you've unlocked a huge chunk of easy points.

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→Linear Functions →Systems of 2 Linear Equations in 2 Variables →Linear Inequalities in 1 or 2 Variables

A linear equation in two variables is just an equation whose graph is a straight line. The two variables are usually x and y. The most useful form to know is slope-intercept form:

y = mx + b

Here's what each letter means:

m is the slope — how steep the line is, or how much y changes every time x goes up by 1. A bigger m means a steeper line.
b is the y-intercept — the value of y when x = 0. On a graph, it's where the line crosses the vertical y-axis.

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y = 3x + 7: the line crosses the y-axis at b = 7, and rises 3 for every 1 step right (slope = 3).

So in y = 3x + 7, the slope is 3 and the y-intercept is 7. The line crosses the y-axis at the point (0, 7).

Intercepts are a big deal on the test:

The y-intercept is where the line hits the y-axis. There, x = 0. Plug in x = 0 and solve for y.
The x-intercept is where the line hits the x-axis. There, y = 0. Plug in y = 0 and solve for x.

That's the key trick: to find an intercept, set the other variable to 0.

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Intercepts: the y-intercept sits on the y-axis (x = 0); the x-intercept sits on the x-axis (y = 0).

Not every equation comes in slope-intercept form. You'll also see standard form:

Ax + By = C

for example 4x + 5y = 20. You can either solve for y to get it into y = mx + b form, or — if you just need an intercept — plug in 0 for one variable.

Word problems are where this really pays off. When a problem has a fixed starting amount plus a steady rate, it's linear. The starting amount is b (the y-intercept) and the rate is m (the slope). A taxi that charges $3.50 to start plus$ 2.25 per mile gives C = 2.25m + 3.50 — rate times miles, plus the flat fee.

The whole game is matching real-world language to m and b:

"flat fee," "starting," "initial," "one-time" → the constant b
"per mile," "each hour," "every," "rate of" → the slope m

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Quick check

Check your understanding with a question from this topic:

A line has the equation y = 3x + 7. What is the y-intercept of this line?

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Worked examples

Example 1

A line has the equation y = -2x + 9. What is the y-intercept of this line?

Example 2

A gym charges a one-time sign-up fee of $40 plus$ 15 per month. Which equation gives the total cost C, in dollars, after m months of membership?

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Example 3

What is the x-intercept of the line 3x - 4y = 24?

Common pitfalls

Confusing slope with y-intercept

In y = mx + b, students often grab the wrong number. The slope m is the coefficient attached to x; the y-intercept b is the lonely constant. Always identify which is which before answering.

Setting the wrong variable to 0 for intercepts

For an x-intercept, set y = 0. For a y-intercept, set x = 0. It feels backwards — to find x you zero out y — but that's exactly the rule. Mixing these up gives the other intercept.

Attaching the rate to the wrong quantity in word problems

A '$15 per month' rate must multiply the months variable, and the one-time fee stays alone. Writing 40m + 15 instead of 15m + 40 is a classic trap — check that the 'per' amount sits next to the matching variable.

Forgetting to solve for y in standard form

Equations like 4x + 5y = 20 aren't in y = mx + b form, so you can't read the slope off directly. Solve for y first, or just plug in 0 if you only need an intercept.

Key takeaways

Slope-intercept form is y = mx + b: m = slope (rate of change), b = y-intercept (value when x = 0).
To find the y-intercept, set x = 0; to find the x-intercept, set y = 0.
In word problems, a flat/one-time amount is the constant b and a 'per-unit' rate is the slope m.
Standard form Ax + By = C can be rearranged into slope-intercept form by solving for y.
The slope is always the number multiplied by x — never the standalone constant.

Watch & learn

Curated Khan Academy walkthroughs on Linear Equations in 2 Variables. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

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Try it yourself

5 practice questions on Linear Equations in 2 Variables, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Linear Equations in 2 Variables

You are here

→Linear Functions →Systems of 2 Linear Equations in 2 Variables →Linear Inequalities in 1 or 2 Variables