SAT System of Equations: Lesson, Practice Problems & Video Explanations

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By Glyn Caddell, BSME · Author of Digital SAT Math Secrets · Tutoring since 2008 — Updated August 18, 2026

System of equations questions ask students to solve for a variable in a system of two equations with two variables or to identify an equation(s) that represents a mathematical relationship in a text. System of equations questions appear on approximately 8.1% of the Digital SAT. System of equations questions fall under Algebra according to the College Board.

Table of Contents

System of Equations Lesson
Tips
Example
- Video Explanation
- Text Explanation
Practice Problems with Video Explanations
Common Mistakes
FAQs
Related Topics

System of Equations Lesson

The lesson covers the types of system of equations questions that show up on the Digital SAT. The lesson covers multiple ways to solve them including graphing, substitution, and elimination.

What Do System of Equations Questions Look Like?

System of Equations questions will typically have two equations, a graph with two equations, or a word problem that asks to identify two equations from the given situation. Here are some examples of System of Equations questions from the Digital SAT.

The solution to the given system of equations is ( $x$ , $y$ ). What is a possible value of $x$ ?
The solution to the given system of equations is ( $x$ , $y$ ). What is the value of $x+y$ ?
The solution to the given system of equations is ( $x$ , $y$ ). What is the value of $x-y$ ?
The solution to the given system of equations is ( $x$ , $y$ ). What is the value of $5x$ ?
The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution ( $x$ , $y$ ) to this system?

1. Graphing Method

You graph both equations on a coordinate plane.
The point(s) where the lines intersect is the solution.
On the SAT, sometimes the graph is given, and you just need to read the intersection point.
You can also use Desmos (https://www.desmos.com/calculator) to quickly graph both equations and see where they cross.

2. Substitution Method

Use this when one equation is already solved (or can easily be solved) for one variable.

Steps:

Solve one equation for one variable (like $y = 2x + 5$ ).
Substitute that expression into the other equation.
Solve for the remaining variable.
Plug back in to find the other variable.

Example:

$y = 4x + 1$ $3x - y = -3$

Substitute $y = 4x + 1$ into the second equation:

$3x - (4x + 1) = -3$

Solve:

$3x - 4x - 1 = -3$ → $-x - 1 = -3$ → $x = 2$

Plug into $y = 4x + 1:$

$y = 4(2) + 1 = 9$

Solution: $(2, 9)$ .

3. Elimination Method

Use this when equations are lined up nicely.

Steps:

Multiply one or both equations so that one variable has the same coefficient.
Add or subtract the equations to eliminate one variable.
Solve for the remaining variable.
Substitute back to find the other variable.

Example:

$3x + 2y = 12$

$5x - 2y = 4$

Add them:

$(3x + 2y) + (5x - 2y) = 12 + 4$ → $8x = 16$ → $x = 2$

Plug into first equation:

$3(2) + 2y = 12$ → $6 + 2y = 12$ → $y = 3$

Solution: $(2, 3)$ .

Tips

If equations are already in slope-intercept form, graph them in Desmos to find the intersection quickly.
Use elimination when coefficients line up nicely.
Use substitution when one equation is already solved for a variable.
Always check if the system has one solution, no solution, or infinitely many solutions (parallel lines = no solution, same line = infinitely many).

Example

A theater sold adult tickets and student tickets for a school performance. Adult tickets cost $14 each, and student tickets cost $9 each. A total of 120 tickets were sold, and the total revenue was $1,430.

How many student tickets were sold?

A) 38
B) 50
C) 62
D) 82

Video Explanation

Text Explanation

Answer: B) 50

Let a be the number of adult tickets sold, and let s be the number of student tickets sold.

Since 120 tickets were sold:

$a+s=120$

Since adult tickets cost $14 and student tickets cost $9:

$14a+9s=1430$

Solve the first equation for $a$ :

$a=120-s$

Substitute $120-s$ for $a$ in the revenue equation:

$14(120-s)+9s=1430$

Distribute:

$1680-14s+9s=1430$

Combine like terms:

$1680-5s=1430$

Subtract 1680 from both sides:

$-5s=-250$

Divide both sides by $-5$ :

$s=50$

Therefore, 50 student tickets were sold.

Common Mistakes

Here are four common mistakes students make when answering system of equations questions.

Finding the Value of the Wrong Variable

A common mistake for system of equations questions is solving for the wrong variable. The equations may be set up to easily solve for $x$ with elimination, so that’s what the student does. However, the question might ask for the value of $y$ .

Not Reading the Prompt Correctly

Another common mistake is not reading the prompt correctly. Sometimes, the question doesn’t ask for $x$ or $y$ . Instead, it asks for something similar to $x+y$ .

Not Using Desmos

Most system of equations questions can be solved quickly using Desmos. Simply enter both equations into Desmos and click on the point of intersection(s). Wherever the equations intersect is the solution(s).

Reorganizing Equations by Solving for y

A common mistake is reorganizing the equations to graph them in Desmos. This isn’t inherently a math mistake, but it’s a test-taking mistake for two reasons:

It wastes time. The equations do not need to be solved for $y$ or in any specific form in order to graph them in Desmos. Simply enter them as they are provided.
It introduces an opportunity for an error. When you try to reorganize an equation, you introduce a chance to make a mistake. There’s no reason to introduce a chance for a mistake when the equations can be graphed as-is.

Frequently Asked Questions

Here are three frequently asked questions that I get from students when reviewing system of equations.

How do you find the solution to a system of equations from a graph?

The solution to a system of equations can be found from a graph by identifying the point(s) of intersection. The set of coordinates of each point of intersection is a solution.

How many solutions does a system of equations have?

Each system of equations can have a different number of solutions.

Two linear functions

Two linear functions can have zero, one, or infinitely many solutions.

They will have zero if they are parallel lines because they will never intersect.

They will have one if they are lines with different slopes. They will intersect once.

They will have infinitely many solutions if they are the same lines. It may not be obvious that they are the same equations until you graph them.

A linear function and a quadratic function

A linear function and a quadratic function can have zero, one, or two solutions.

Do I have to know substitution or elimination for the Digital SAT?

No, you do not need to know substitution or elimination to solve a system of equations for the Digital SAT. A system of equations can be solved using Desmos. However, there are times when knowing how to do substitution or elimination can save you time on the test.