Algebra · SAT Math

Systems of Equations on the Digital SAT

A system of equations is just two (or more) equations that share the same variables — and the SAT tests them heavily. This lesson covers both classical methods (substitution and elimination), the no-solution / infinite-solutions rules that show up in "find k" questions, word problems, and 8 practice problems with full solutions.
By the Prepiii Editorial TeamUpdated 2026-05-23~11 min read

What is a system of equations?

A system is two equations that both involve the same variables — usually x and y. The solution is the pair (x, y) that makes both equations true at the same time. Graphically, the solution is the point where the two lines intersect.

The SAT mostly tests 2-variable, 2-equation systems where both equations are linear. Two main solution methods: substitution and elimination.

Method 1: Substitution

Use substitution when one equation is already solved for a variable (or easy to solve). The recipe:

  1. Solve one equation for one variable.
  2. Substitute that expression into the other equation.
  3. Solve for the remaining variable.
  4. Plug back to find the other variable.

Example. Solve the system y = 2x + 1 and 3x + y = 11.

Substitute y = 2x + 1 into the second equation: 3x + (2x + 1) = 11.

Simplify: 5x + 1 = 11 → 5x = 10 → x = 2.

Plug back: y = 2(2) + 1 = 5. Solution: (2, 5).

Method 2: Elimination

Use elimination when neither equation is solved for a variable — usually when both equations are in the form ax + by = c. The idea: add (or subtract) the equations so one variable cancels.

  1. Multiply one or both equations so the coefficient of one variable matches in size (and opposite in sign).
  2. Add the equations to eliminate that variable.
  3. Solve for the remaining variable.
  4. Plug back to find the other variable.

Example. Solve 2x + 3y = 12 and 4x - 3y = 6.

Add the equations directly — the 3y and -3y cancel: 6x = 18 → x = 3.

Plug back: 2(3) + 3y = 12 → 3y = 6 → y = 2. Solution: (3, 2).

No, one, or infinite solutions

A system of two linear equations always has exactly one of three outcomes:

One solution. The two lines have different slopes and cross at exactly one point. (Default case.)

No solution. The lines are parallel — same slope, different y-intercepts. They never meet.

Infinitely many solutions. The two equations are the same line in disguise — same slope, same y-intercept. Every point on the line satisfies both.

SAT pattern: "find k" questions. The test loves to give you a system with an unknown k and ask which value of k makes the system have no solution (or infinite solutions). Set the slopes equal, then check the y-intercepts.

Word problems with systems

The recipe for system word problems is always: two unknowns means two equations. Look for two distinct pieces of information that each constrain the unknowns.

  1. Name the two unknowns. Write what each variable means.
  2. Translate each sentence into an equation.
  3. Solve by substitution or elimination.
  4. Answer the actual question (it may ask for just one of the variables, or a combination).

Common scenarios: ticket sales (adult + child prices), mixtures (two solutions of different concentrations), age problems, money problems, and distance/time/rate.

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Solving with Desmos

Systems are where Desmos saves the most time on the SAT. Three techniques cover almost everything:

1. Graph both equations, click the intersection

Type both equations on separate lines. Desmos shows where they cross with a grey dot. Click the dot to see the exact (x, y) coordinates. That's your solution.

2. No-solution & find-k — use a slider

For "find k such that the system has no solution": type both equations with k in place. Desmos prompts to add a slider — drag it until the two lines become parallel (no intersection). Read the value of k.

3. Word problems — verify your equations

Once you've translated the word problem into a system, type it into Desmos and click the intersection. If the answer matches an answer choice, you set up the equations correctly. If not, re-read the problem.

For the full set of Desmos techniques across the entire test, see our Desmos & Test Tools guides.

Common mistakes

Substituting into the same equation you solved

If you solved Equation A for y, you must substitute into Equation B — not back into A (you'll just get y = y, a tautology).

Forgetting to find both variables

Solving for x is half the work. Plug back in to find y too — the SAT often asks for the product xy or the sum x + y, which requires both.

Confusing 'no solution' with 'one solution = 0'

x = 0 is a solution. 'No solution' means there is no x that satisfies the system at all. Different things.

Setting up word-problem equations from the wrong totals

If 100 tickets sold for a total of $750 with adult ($10) and child ($5) tickets, the equations are a + c = 100 and 10a + 5c = 750. Mixing up which is the count and which is the dollars is the #1 setup error.

Practice problems

8 problems adapted from College Board released questions and internal Prepiii sets. Click each one to reveal the solution.

1

What is the solution (x, y) to the system y = x + 3 and 2x + y = 12?

  1. (2, 5)
  2. (3, 6)
  3. (4, 7)
  4. (5, 8)

Click to reveal solution →

Answer: (B) (3, 6)

Substitute y = x + 3 into the second equation: 2x + (x + 3) = 12 → 3x = 9 → x = 3. Then y = 3 + 3 = 6.

2

For the system 3x + 2y = 12 and 5x - 2y = 4, what is the value of x?

  1. 1
  2. 2
  3. 3
  4. 4

Click to reveal solution →

Answer: (B) 2

Add the equations: the 2y and -2y cancel, leaving 8x = 16, so x = 2.

3

For what value of k does the system y = 2x + 5 and y = kx - 1 have no solution?

  1. -2
  2. 0
  3. 2
  4. 5

Click to reveal solution →

Answer: (C) 2

No solution means the lines are parallel — same slope, different y-intercept. Slope of the first line is 2, so k = 2. The y-intercepts (5 vs -1) are already different, so parallel is achieved.

4

A movie theater sells adult tickets for $12 and child tickets for $8. On Friday night they sold 100 tickets for a total of $1,040. How many adult tickets did they sell?

Click to reveal solution →

Answer: 60

Let a = adult tickets, c = child tickets. Then a + c = 100 and 12a + 8c = 1040.

From the first: c = 100 - a. Substitute: 12a + 8(100 - a) = 1040 → 4a + 800 = 1040 → a = 60.

5

The system 4x + 6y = 12 and 2x + 3y = 6 has how many solutions?

  1. None
  2. Exactly one
  3. Exactly two
  4. Infinitely many

Click to reveal solution →

Answer: (D) Infinitely many

Divide the first equation by 2: 2x + 3y = 6. That's the same as the second equation. Same line in disguise → infinite solutions.

6

If (x, y) is a solution to x - y = 4 and 2x + y = 11, what is the value of x + y?

  1. 4
  2. 6
  3. 8
  4. 11

Click to reveal solution →

Answer: (B) 6

Add the equations: 3x = 15 → x = 5. Plug back: 5 - y = 4 → y = 1.

x + y = 5 + 1 = 6.

7

The system y = 3x + 7 and y = 3x + 2 has how many solutions?

  1. None
  2. Exactly one
  3. Exactly two
  4. Infinitely many

Click to reveal solution →

Answer: (A) None

Both lines have slope 3, but different y-intercepts (7 vs 2). They're parallel — never meet.

8

A 30-liter solution contains 20% acid. How many liters of pure acid must be added to make the solution 50% acid?

  1. 12
  2. 15
  3. 18
  4. 20

Click to reveal solution →

Answer: (C) 18

Pure acid currently: 0.20 × 30 = 6 liters. Let x = liters of pure acid added.

New acid amount: 6 + x. New total: 30 + x. Set up: (6 + x) / (30 + x) = 0.50.

Solve: 6 + x = 0.5(30 + x) → 6 + x = 15 + 0.5x → 0.5x = 9 → x = 18.

Frequently asked questions

What is a system of equations?

+
A system of equations is two or more equations that share the same variables. The solution is the set of values that makes all the equations true simultaneously. On the SAT, you'll usually deal with systems of two linear equations in two variables.

Substitution or elimination — which method is faster?

+
Use substitution when one equation is already solved for a variable (like y = 2x + 1). Use elimination when both equations are in the form ax + by = c. With Desmos, you can often skip both methods and just read the intersection point.

When does a system have no solution?

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When the two lines are parallel — same slope, different y-intercepts. They never intersect, so no (x, y) pair satisfies both equations.

When does a system have infinitely many solutions?

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When the two equations describe the same line. One is just a scalar multiple of the other. Every point on the line satisfies both equations.

How do I solve word problems with systems?

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Identify the two unknowns, then translate each sentence into an equation. Two unknowns require two distinct equations. Solve by substitution or elimination, then answer what the question actually asks (which may be a combination like x + y or 2x).

Keep going

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The foundation for understanding systems

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