Digital SAT · Solve it in Desmos

How to Solve Systems of Linear Equations on Desmos (Digital SAT)

Desmos solves nearly every graphable system and removes arithmetic slips. Reach for algebra by hand when the target is a sum of the equations you can combine directly, or on the number-of-solutions special cases where there is no point to click.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for systems of linear equations

The Desmos steps

1
Type both equations
Enter each on its own line, exactly as written, in any form; Desmos accepts them as-is.
2
Click the intersection
The grey dot where the lines cross is the solution; click it for the coordinates.
3
Answer the exact quantity
Read x, y, the pair, or the requested combination, computing a sum in Desmos if needed.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Easy example

The system of equations

5x = 15

and

-4x + y = -2

has solution

(x, y)

. What is the value of

x + y

-17

-13

13

17

A: Incorrect. This applies sign errors to both $x$ and $y$ .

B: Incorrect. This negates the correct sum.

C: Correct. $x = 3$ , and $-12 + y = -2$ gives $y = 10$ , so $x + y = 13$ .

D: Incorrect. This uses $y = 14$ from a sign slip on $-2$ .

Explanation

From $5x = 15$ , $x = 3$ . Then $-4(3) + y = -2$ gives $y = 10$ . So $x + y = 13$ .

Medium example

One equation in a system of two linear equations is

y = \frac{2}{7}x + 3

. The system has infinitely many solutions, and the second equation is

y = mx + b

, where

m

and

b

are constants. What is the value of

b

-3

-\frac{1}{3}

\frac{1}{3}

3

A: Incorrect. This negates the intercept.

B: Incorrect. This inverts and negates the intercept.

C: Incorrect. This inverts the intercept.

D: Correct. Infinitely many solutions means the equations are identical, so $b = 3$ .

Explanation

Infinitely many solutions means the two equations are the same line, so $b$ must equal the given intercept, 3.

Hard example

The solution to the system of equations

5x + 2y = 16

and

3x + 2y = 8

(x, y)

. What is the value of

8x + 4y

8

16

24

48

A: Incorrect. This uses only the second equation's value.

B: Incorrect. This uses only the first equation's value.

C: Correct. Adding the equations gives $8x + 4y = 16 + 8 = 24$ .

D: Incorrect. This doubles the correct sum.

Explanation

Adding the two equations directly: $(5x + 2y) + (3x + 2y) = 16 + 8$ , so $8x + 4y = 24$ .

Detailed explanation

Notice that $8x + 4y$ is exactly the sum of the two left sides: $(5x + 2y) + (3x + 2y) = 8x + 4y$ . Adding the right sides gives $16 + 8 = 24$ , so $8x + 4y = 24$ without solving for $x$ and $y$ separately.

Try it with the calculator

Question 1easy

The system of equations $s + 7r = 27$ and $r = 3$ has solution $(r, s)$ . What is the solution $(r, s)$ ?

Question 2easy

The system of equations $x = 5$ and $y = x - 8$ has solution $(x, y)$ . Which point is the solution?

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

Desmos questions

Can Desmos solve systems of linear equations questions on the SAT?

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master systems of linear equations, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

See the full method Take the free diagnostic

The Desmos steps

Type both equations

Click the intersection

Answer the exact quantity

Worked in Desmos

Try it with the calculator

Question 3 is ready when you are

Desmos questions

Can Desmos solve systems of linear equations questions on the SAT?

Is the Desmos calculator really built into the Digital SAT?

Should I still learn the algebra?

Master systems of linear equations, calculator and all