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Systems of Two Linear Equations for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about systems of linear equations in one place: solving by substitution, elimination, and graphing, the number of solutions a system can have, and word problems — with step-by-step examples, worked problems, and a free practice quiz.
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Free · No signupSystems of Two Linear Equations
Substitution, elimination, graphing, and number of solutions.
What is a system of linear equations?
A system of two linear equations is a pair of equations involving the same two variables, considered together. Solving the system means finding the (x, y) pair — if one exists — that makes both equations true at the same time.
Geometrically, each linear equation represents a line. Solving the system means finding where those two lines intersect — which is exactly why a system can have one, zero, or infinitely many solutions.
Solving by substitution
Substitution works best when one equation is already solved for a variable, or can be solved for one easily.
Solve the system: y = 2x + 1 and 3x + y = 16
Solving by elimination
Elimination works by adding or subtracting the two equations to cancel out one variable entirely, often after multiplying one or both equations by a constant first.
Solve the system: 2x + 3y = 16 and 2x − y = 4
If neither variable cancels immediately, multiply one or both equations by a constant so that the coefficients of one variable become opposites (or identical), then add or subtract.
Solving by graphing
Graphing both equations and identifying the intersection point gives the solution directly. This method is most useful when the PSAT shows you a graph, or when the equations are already in a graph-friendly form like slope-intercept.
When the PSAT provides a graph of a system instead of equations, you don't need to solve algebraically at all — simply read the coordinates of the intersection point directly from the graph.
How many solutions can a system have?
| Line relationship | Number of solutions | Slope & intercept |
|---|---|---|
| Lines intersect at one point | Exactly one solution | Different slopes |
| Lines are parallel | No solution | Same slope, different y-intercept |
| Lines are identical | Infinitely many solutions | Same slope, same y-intercept |
Does the system y = 4x + 7 and y = 4x − 2 have a solution?
A system with "no solution" or "infinitely many solutions" question type often asks you to find a missing coefficient that makes the slopes equal, then separately check whether the intercepts also match. Don't stop after matching just the slope.
System of equations word problems
A movie theater sells adult tickets for $12 and child tickets for $8. One night, 150 tickets were sold for a total of $1,560. How many adult tickets were sold?
Common PSAT traps
- Forgetting to back-substitute: after finding one variable, always plug it back in to find the second.
- Sign errors during elimination: subtracting equations means subtracting every term, including the ones with matching signs.
- Stopping after matching only the slope: for "no solution" vs. "infinite solutions" questions, check both slope and y-intercept.
- Mismatched units in word problems: double-check that both equations use consistent units before solving.
Test day strategy for systems of linear equations
| Question signal | Fastest approach |
|---|---|
| One equation already solved for a variable | Use substitution |
| Both equations in standard form with matching coefficients | Use elimination |
| Graph of the system provided | Read the intersection point directly — no algebra needed |
| "No solution" or "infinitely many solutions" | Compare slopes first, then y-intercepts |
| Word problem describing two related quantities | Define two variables, write two equations, then solve |
Now put it to work
One comprehensive quiz covering the full topic — substitution, elimination, graphing, and the number of solutions.
Systems of Two Linear Equations
Substitution, elimination, graphing, and number of solutions.