PSAT/NMSQT Math: Systems of Two Linear Equations
A system of two linear equations asks you to find the single (x, y) pair, if one exists, that satisfies both equations simultaneously. This free, complete lesson covers solving systems by substitution, solving by elimination, translating word problems into a system of equations, and reading a system's solution directly from a graph. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Solving by Substitution
Substitution works best when one equation is already solved for a variable, or can easily be solved for one. Substitute that expression into the other equation, solve for the remaining variable, then plug back in to find the first variable.
Worked Example: Solve the system: y = 2x + 1 and 3x + y = 16.
Substitute y = 2x + 1 into the second equation:
3x + (2x + 1) = 16 → 5x = 15 → x = 3, so y = 2(3) + 1 = 7
1. Solve the system: y = x + 4 and 2x + y = 19
Explanation: 2x + (x + 4) = 19 → 3x = 15 → x = 5, so y = 5 + 4 = 9.
2. Solve the system: x = 3y − 2 and 4x + y = 31
Explanation: 4(3y − 2) + y = 31 → 12y − 8 + y = 31 → 13y = 39 → y = 3, so x = 3(3) − 2 = 7.
3. Solve the system: y = −x + 6 and y = 2x − 3
Explanation: −x + 6 = 2x − 3 → 9 = 3x → x = 3, so y = −3 + 6 = 3.
2. Solving by Elimination
Elimination works by adding or subtracting the two equations so that one variable cancels out. If the coefficients of a variable aren't already opposites, multiply one or both equations by a constant first so that they become opposites (or matching, for subtraction).
Worked Example: Solve the system: 2x + 3y = 16 and 2x − y = 4.
Subtract the second equation from the first to eliminate x:
4y = 12 → y = 3, and 2x − 3 = 4 → x = 3.5
1. Solve the system: x + y = 10 and x − y = 2
Explanation: Adding both equations: 2x = 12 → x = 6, so y = 10 − 6 = 4.
2. Solve the system: 3x + 2y = 12 and x + 2y = 8
Explanation: Subtracting: 2x = 4 → x = 2, so 2 + 2y = 8 → y = 3.
3. Solve the system: 2x + y = 9 and 3x + 2y = 16
Explanation: Multiply the first equation by 2: 4x + 2y = 18. Subtract the second equation: x = 2, so 2(2) + y = 9 → y = 5.
3. Writing Systems from Word Problems
Word problems that describe two unknown quantities related by two separate conditions translate into a system of two equations. Identify what each variable represents, then write one equation per condition given in the problem.
Worked Example: A movie theater sells adult tickets for $12 and child tickets for $8. One showing sold 150 tickets total for $1,560. Write a system to find the number of adult tickets a and child tickets c sold.
a + c = 150 and 12a + 8c = 1560
1. A store sells pens for $2 and notebooks for $5. A customer buys 8 items total for $28. Which system represents this, using p for pens and n for notebooks?
Explanation: Total items sold is 8 (p + n = 8), and total cost is 28, with pens at $2 and notebooks at $5 (2p + 5n = 28).
2. Using the system p + n = 8 and 2p + 5n = 28 from the previous problem, how many notebooks, n, were bought?
Explanation: From p + n = 8, p = 8 − n. Substituting: 2(8 − n) + 5n = 28 → 16 + 3n = 28 → n = 4.
3. The sum of two numbers is 24, and their difference is 6. What are the two numbers?
Explanation: Let the numbers be x and y: x + y = 24 and x − y = 6. Adding: 2x = 30 → x = 15, so y = 9.
4. Reading Systems from a Graph
Graphically, each equation in a system is a line, and the solution to the system is the point where the two lines intersect. If two lines are parallel (same slope, different y-intercepts), the system has no solution. If the two equations describe the exact same line, the system has infinitely many solutions.
Worked Example: Two lines on a graph cross at the point (2, 5). What is the solution to the system of their equations?
The intersection point itself is the solution:
x = 2, y = 5
1. A graph shows the lines y = 3x − 1 and y = 3x + 4. How many solutions does this system have?
Explanation: Both lines have slope 3 but different y-intercepts (−1 and 4), so they are parallel and never intersect, meaning no solution.
2. A system consists of y = 2x + 5 and 2y = 4x + 10. How many solutions does this system have?
Explanation: Dividing the second equation by 2 gives y = 2x + 5, identical to the first equation, so every point on the line is a solution.
3. On a graph, one line passes through (0, 2) and (4, 10), and another passes through (0, 8) and (4, 0). Based only on their y-intercepts and general direction, what can you say about their intersection?
Explanation: The first line rises (slope 2) while the second falls (slope −2); since their slopes differ, two distinct lines must cross at exactly one point.
Common Mistakes to Avoid
- Forgetting to distribute a negative sign when subtracting equations in elimination. Every term on the right side of the equation being subtracted must also flip sign.
- Solving for only one variable and forgetting to find the second. A system's full solution is an (x, y) pair, not a single number.
- Mismatching which quantity goes with which coefficient in a word problem. Carefully match each condition in the problem to a full, separate equation before solving.
- Assuming every system has exactly one solution. Parallel lines produce no solution, and identical lines produce infinitely many, both of which are tested directly on the PSAT.
- Multiplying only one side of an equation when preparing to eliminate a variable. Every term on both sides must be multiplied by the same constant.
Frequently Asked Questions
How do I decide whether to use substitution or elimination?
Use substitution when one equation is already solved (or easily solved) for a variable. Use elimination when the equations are in standard form (Ax + By = C) and one variable's coefficients are the same or easy to match by multiplying.
How can I tell a system has no solution without graphing it?
Write both equations in slope-intercept form. If the slopes are equal but the y-intercepts are different, the lines are parallel and the system has no solution.
Where can I practice more problems like these?
The Systems of Two Linear Equations quiz in the PSAT/NMSQT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.