Digital SAT · Linear Functions

How to Solve Intercepts on the Digital SAT

An intercept is where a line meets an axis. For the x-intercept, set y to zero and solve for x; for the y-intercept, set x to zero and solve for y, or just read the constant in slope-intercept form. On the digital test the quickest path is to graph the line and click each axis crossing, which Desmos labels. The one thing to guard is which coordinate the question actually wants.

Written from Perfect1600’s analysis of every intercepts question in our bank·Method checked against the current Bluebook test

per test: 1 per test
typical difficulty: Mostly easy
practice questions: 38

How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

Jump to free practice ↓

What the question bank shows

x=0 / y=0

Zero the other variable

Set x to zero for the y-intercept, y to zero for the x-intercept.

55%

Click it on Desmos

Graph the line and click where it crosses each axis.

the trap

Axis swap

Reversing which variable to zero out is the frequent slip.

How to recognize intercepts questions

A line is given by an equation or by two points, and an intercept is asked.
It wants the x-intercept, the y-intercept, or a coordinate where the line meets an axis.
The equation may be in standard, slope-intercept, or fraction form.
Choices are numbers or points, and many are grid-ins.

Why students miss these

The rule is short, so the errors are axis swaps and messy forms. Setting x to zero gives the y-intercept, not the x-intercept, and under time pressure that gets reversed. Fraction and standard forms bring in algebra where a sign or a division slips, and on a grid-in there is no choice to flag it. Graphing skips both problems by showing the crossings directly.

The step-by-step method

1
Decide which intercept
The x-intercept is on the x-axis, where y = 0; the y-intercept is on the y-axis, where x = 0.
2
Set the other variable to zero
For the x-intercept, plug in y = 0 and solve for x; for the y-intercept, plug in x = 0 and solve for y.
3
Or read it from the form
In y = mx + b the y-intercept is b; from two points, it is where the line hits x = 0.
4
Answer the right coordinate
Give the value or point wanted, and check you did not hand back an x where a y was asked.

Solving it on Desmos

Graph the equation. Type the line in any form to see it plotted.
Click the crossings. The x-axis crossing for the x-intercept, the y-axis crossing for the y-intercept; each is labeled.
Take the value asked. Read the coordinate the question wants from the labeled crossing.

Full Desmos walkthrough for intercepts→

When to use it: Graphing is quickest for fraction or standard forms where the algebra is fiddly, and it removes the axis-swap error. Read by hand only when the equation is already in slope-intercept form.

See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Medium example

A line passes through the points

(0, -3)

and

(2, 5)

in the xy-plane. What is the y-coordinate of the y-intercept of the line?

-3

1

4

5

A: Correct. The point $(0, -3)$ is on the y-axis, so the y-intercept is $-3$ .

B: Incorrect. This averages the two y-values instead of reading $x = 0$ .

C: Incorrect. This is the slope of the line, not the y-intercept.

D: Incorrect. This is the y-value at $x = 2$ , not at $x = 0$ .

Explanation

The y-intercept is where $x = 0$ ; the point $(0, -3)$ gives $y = -3$ .

Medium example

The equation

5x + 2y = 20

represents a line in the xy-plane. What is the x-coordinate of the x-intercept of the line?

-4

4

10

20

A: Incorrect. This has the wrong sign for the x-intercept.

B: Correct. Setting $y = 0$ gives $5x = 20$ , so $x = 4$ .

C: Incorrect. This is the y-coordinate of the y-intercept, not the x-intercept.

D: Incorrect. This uses the constant 20 without dividing by 5.

Explanation

The x-intercept occurs at $y = 0$ : $5x = 20$ , so $x = 4$ .

Hard example

A line in the xy-plane is defined by

\frac{x}{5} - \frac{y}{2} = 1

. What is the x-coordinate of the x-intercept of the line?

-2

1

2

5

A: Incorrect. This is the y-coordinate of the y-intercept, not the x-intercept.

B: Incorrect. This is the right side of the equation, not the intercept.

C: Incorrect. This uses the denominator under $y$ instead of under $x$ .

D: Correct. Setting $y = 0$ gives $\frac{x}{5} = 1$ , so $x = 5$ .

Explanation

The x-intercept occurs at $y = 0$ : $\frac{x}{5} = 1$ , so $x = 5$ .

Detailed explanation

Setting $y = 0$ leaves $\frac{x}{5} = 1$ , so $x = 5$ . The denominator under $x$ gives the x-intercept directly.

The common traps

Pattern	What it does	The tell
Axes swapped	Set x = 0 when the x-intercept was asked, or the reverse.	The x-intercept needs y = 0; the y-intercept needs x = 0.
Wrong coordinate	Gave the y-value of an intercept where the x-value was asked.	Check which coordinate the question wants.
Sign or division slip	Mishandled a negative or fraction in standard or fraction form.	Recheck the arithmetic, or graph and click to confirm.
b read as x-intercept	Took the constant term as the x-intercept.	In y = mx + b, b is the y-intercept; the x-intercept needs y = 0.

Try it: two real questions

Question 1hard

A line in the xy-plane has a slope of 3 and passes through the point $(2, 5)$ . What is the y-coordinate of the point where the line crosses the y-axis?

Question 2hard

A line in the xy-plane has a slope of $-2$ and passes through the point $(3, 4)$ . What is the x-coordinate of the x-intercept of the line?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

Start the free diagnostic

Often confused with

Linear Functions

Slope of a Line

Linear Functions

Slope and Intercept in Context

Linear Functions

Parallel and Perpendicular Lines

Common questions

How do you find the x-intercept of a line on the SAT?

Set y to zero and solve for x. That is where the line crosses the x-axis. Or graph the line and click the x-axis crossing to read it exactly.

How do you find the y-intercept?

Set x to zero and solve for y, or read the constant b in y = mx + b. In Desmos, click where the line crosses the y-axis.

What is the difference between the x-intercept and the y-intercept?

The x-intercept is on the x-axis, where y = 0; the y-intercept is on the y-axis, where x = 0. Swapping them is the most common mistake.

Can Desmos find intercepts for me?

Yes. Type the equation in any form and click each axis crossing; Desmos labels the coordinates, which avoids the axis-swap and arithmetic errors.

How do I find an intercept from two points?

Find the line's equation first, then set the appropriate variable to zero. Or plot both points in Desmos and click the intercepts.

Practice intercepts the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

Start the free diagnostic See the score guarantee

6,200+ tagged questions · every question type analyzed · money-back score guarantee

What the question bank shows

How to recognize intercepts questions

Why students miss these

The step-by-step method

Decide which intercept

Set the other variable to zero

Or read it from the form

Answer the right coordinate

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do you find the x-intercept of a line on the SAT?

How do you find the y-intercept?

What is the difference between the x-intercept and the y-intercept?

Can Desmos find intercepts for me?

How do I find an intercept from two points?

Practice intercepts the way it is tested