Solve systems of two linear equations, interpret their solutions and decide when a system has none or infinitely many. It is part of Algebra in the Math section. The book teaches it in Chapter 18, Systems of Two Linear Equations, with 16 practice questions, and tests it 8 more times across the two full-length practice tests.
About Chapter 18
A system of two linear equations is a pair of linear equations in the same two variables. A solution to the system is an ordered pair (x, y) that makes both equations true at the same time; on a graph, it is the point where the two lines meet. Systems questions can appear in either Math module, as stacked equations, word problems or graphs, in multiple-choice or student-produced response (SPR) form.
This chapter gives you two algebra methods, a quick test for the number of solutions and a routine for turning a story into a system, with the graphing calculator as your check.
What the question looks like
Systems questions come in four families.
Solve; Typical stem: What is the solution (x, y) to the given system of equations?; What you produce: x, y, or an expression such as x + y
Number of solutions; Typical stem: How many solutions does the given system have? The given system has no solution. What is the value of k?; What you produce: zero, one or infinitely many, or a constant
Context; Typical stem: Which system of equations represents this situation? How many adult tickets were sold?; What you produce: two equations, or one unknown amount
Graph; Typical stem: The graphs of the equations in a system are shown. What is the solution?; What you produce: the intersection point, or what it means
The equations are usually printed one above the other and called "the given system." Context questions often print no equations at all: you build them from two facts in the story.
How to recognize it
Look for any of these signals:
Two equations that use the same two variables, printed together.
The phrases "solution (x, y)," "intersection," "no solution" or "infinitely many solutions."
A story with two unknown amounts and two separate facts about them, usually a total count and a total value (dollars, liters, points).
Two lines drawn in the same xy-plane.
Common traps
Answering with x when the question asks for y, x + y or a constant: Reread the final sentence of the stem before you choose.
Losing a sign when subtracting equations: Subtract every column, constants included, and write the subtraction out.
Scaling only part of an equation: After multiplying, every term in that equation, constant included, should have changed.
Mixing up "no solution" and "infinitely many": No solution: same slope, different intercepts. Infinitely many: the very same line.
Pairing a price or rate with the wrong variable: Build a count row and a value row; each price sits under its own item.
Reading an intercept instead of the intersection: The solution is where the lines cross each other, not where either crosses an axis.
A pair that fits only one equation: Check your answer in both equations before you move on.
Video lessons
Two short lessons from independent YouTube creators, chosen for this skill. The notes are ours; the videos belong to their creators, who are not affiliated with this site.
Solving systems of linear equations (Marcus Maths, 11:39). Watch for: Elimination, substitution and the conditions for no solution or infinitely many.
Solving a system with the graphing calculator (Erika | Math + Engineering, 9:46). Watch for: Graphing both lines and reading the intersection point.
Practice this skill
The full method, worked examples and 16 practice questions with every answer choice explained are in Chapter 18 of the book and in the study workspace, where an AI coach can talk you through each one.
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