Educerie · SAT · Math
Algebra · ALG.4 Systems of two linear equations in two variables
What you must be able to do
| You must be able to | What it looks like on the test |
|---|---|
| Solve by substitution | When one variable is already alone, or easy to isolate |
| Solve by elimination | When neither variable is alone |
| Say how many solutions a system has | One, none, or infinitely many |
| Build a system from a word problem | Two sentences, two equations |
| Solve for an expression like x + y | Often faster than solving for each |
1The idea in one paragraph
Two lines in a plane either cross once, never cross, or are the same line. That is the whole theory: one solution, no solution, infinitely many. Solving means finding where they cross, and the test's favourite question is not the crossing point at all but the count — which you answer by comparing slopes, not by solving. Different slopes cross once; the same slope with different intercepts never crosses; the same slope and the same intercept is one line written twice.
When the answer choices are the number of solutions, put both equations in y = mx + b or compare coefficients. Do not solve.
2Choosing a method
Substitution when one equation already has a variable alone, or a coefficient of 1. y = 2x − 5 goes straight into the other equation.
Elimination when neither does. Line the equations up, scale one or both so a variable's coefficients match, then add or subtract.
Elimination is faster than students expect and is the method to default to when both equations are in standard form.
3How many solutions, without solving
Write both as ax + by = c.
| Condition | Solutions |
|---|---|
| a₁/a₂ ≠ b₁/b₂ | One — different slopes |
| a₁/a₂ = b₁/b₂ but ≠ c₁/c₂ | None — parallel |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Infinitely many — same line |
For what value of k does 6x + ky = 10 and 9x + 12y = 15 have infinitely many solutions?
Half the "no solution" questions on the test are that middle row in disguise.
4Word problems: two sentences, two equations
Twelve items were bought, some at 40 lira and some at 65 lira, for 605 lira in total.
SAT numbers come out clean. If yours do not, check the translation before you check the arithmetic — a decimal answer to a question about items almost always means one of the two equations is wrong.
Always write what each letter means. Two of the four answer choices are usually the other variable's value.
5Solving for an expression
If x + 3y = 14 and 2x − y = 7, what is the value of 3x + 2y?
No solving needed.
Whenever a question asks for a combination rather than for x or y, try adding or subtracting the equations first. When it works it saves a minute.
6The calculator solves any of them
Put both equations in y = form and graph. The intersection is the solution, parallel lines show no crossing, and a single visible line means infinitely many. For a messy system this is faster and safer than elimination.
Where points are lost
- Answering with the wrong variable. Write down which is which, and check what was asked.
- Solving when the question wanted the number of solutions.
- Sign errors in elimination — subtracting a negative term.
- Scaling only part of an equation when multiplying to match coefficients.
- Forgetting to find the second variable when the question needs both.
- Ignoring the ratio test and guessing at parallel.
Work it right
- Read what is wanted: x, y, a combination, or a count.
- If it is a count, compare coefficients and stop.
- Otherwise pick a method: substitution if a variable is alone, elimination if not.
- Solve for one, substitute back for the other.
- Check both equations with your pair before answering.
Try it
Q1. If 4x + 3y = 30 and 2x − 3y = 6, what is the value of x?
A) 2 B) 3 C) 5 D) 6
Q2. The system 5x − 2y = 11 and 15x − 6y = 30 has how many solutions?
A) Zero B) Exactly one C) Exactly two D) Infinitely many
Q3. For what value of c does the system 3x + 4y = 9 and 6x + 8y = c have infinitely many solutions?
A) 9 B) 12 C) 18 D) No value of c
Q4. A cinema sold 240 tickets for 3,720 lira. Adult tickets cost 20 lira and child tickets 12 lira. How many adult tickets were sold? (Type your answer.)
Q5. If a + 2b = 13 and 3a − 2b = 7, what is the value of 4a?
A) 5 B) 10 C) 20 D) 26
In one breath
Two lines cross once, never, or everywhere, and the test asks which far more often than it asks where. Compare the coefficients: different ratios of x and y coefficients mean one solution, equal ratios with a different constant mean none, and equal all the way through means the same line twice. When you do have to solve, use substitution if a variable is already alone and elimination if it is not — and check which letter the question actually wanted.
Answers
Q1. D — 6. add the equations to eliminate y
A is y. B and C come from subtracting the equations instead of adding, which eliminates nothing.
Q2. A — Zero. same slope, different constant Multiplying the first by 3 gives 15x − 6y = 33, which contradicts 15x − 6y = 30. The lines are parallel. D is the trap for a student who sees 15 = 3 × 5 and 6 = 3 × 2 and stops before checking the constant.
Q3. C — 18. every coefficient must scale by the same factor
A keeps the constant unchanged; B is unrelated; D ignores that a consistent scaling exists.
Q4. 105. two equations, then substitute
Typing 135 is the error the question is built for: a correct value of the wrong variable.
Q5. C — 20. add the equations
A is a. B and D come from solving for b or adding the constants.
Educerie · written from the published College Board* Assessment Framework for the Digital SAT Suite *(Math, Algebra, skill/knowledge testing point "Systems of two linear equations in two variables"). All questions and explanations are original Educerie text. Last reviewed 12 September 2026.
Check your understanding
The main ideas of this note. Tick each one you could do now, in an exam, without looking back up. Anything you cannot tick yet is the part to read again.