SAT Math · Heart of Algebra
Systems of two linear equations
Solving 2-by-2 systems by substitution, elimination, or graphing.
What's tested in this subtopic
A staple of every SAT form. The system may be presented in standard form, in slope-intercept form, or as a word problem you must translate. Some items ask only for one variable's value; others ask for the sum, product, or difference of the two variables. A small fraction of items ask for the number of solutions (one, none, or infinitely many) without solving.
Tactics that actually move your score
Decide substitution vs. elimination before you start algebra. If a coefficient is already 1 or −1, substitution is usually faster. If both equations are in standard form, elimination is faster. To check for the number of solutions, divide the coefficients: equal ratio of x and y but different constants means parallel lines (no solution); equal ratio of all three means the same line (infinitely many). The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy Consider the system of equations: 1x + 5y = 31 5x − 4y = 10 What is the value of x?
- Easy Consider the system of equations: 4x + 3y = -27 1x − 4y = 17 What is the value of x?
- Medium Consider the system of equations: 5x + 4y = -31 5x − 4y = 1 What is the value of x?
- Medium Consider the system of equations: 2x + 5y = -12 4x − 1y = 20 What is the value of x?
- Medium Consider the system of equations: 5x + 4y = 46 3x − 2y = 10 What is the value of x?
- Hard Consider the system of equations: 5x + 3y = -2 3x − 3y = -6 What is the value of x?
- Easy Consider the system of equations: 1x + 3y = 3 4x − 1y = -27 What is the value of x?
- Easy Consider the system of equations: 1x + 4y = 17 1x − 2y = -13 What is the value of x?
- Medium Consider the system of equations: 5x + 2y = 12 4x − 1y = 20 What is the value of x?
- Medium Consider the system of equations: 4x + 2y = 24 5x − 1y = 9 What is the value of x?
- Medium Consider the system of equations: 4x + 3y = 21 5x − 4y = 3 What is the value of x?
- Hard Consider the system of equations: 2x + 2y = -14 3x − 3y = -15 What is the value of x?
- Easy Consider the system of equations: 1x + 3y = -3 5x − 4y = -34 What is the value of x?
- Easy Consider the system of equations: 1x + 5y = -24 1x − 5y = 26 What is the value of x?
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in SAT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.