What the exam asks
| Question shape | Typical stem | Usual difficulty |
|---|---|---|
| Solve | “If is the solution to the system, what is the value of ?” | Easy–medium (MCQ or SPR) |
| Combination | “What is the value of ?” or “of ?” |
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Lesson 4 of 5 · Algebra
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Systems of two linear equations are among the most-tested skills in SAT Math. Expect about 2–4 per test, spread over both modules, as bare equation pairs, word problems, graphs and “no solution / infinitely many solutions” parameter questions. The parameter questions are a favorite of the hard module 2. Learn three solving methods and one ratio test, and these become some of the fastest points on the test.
| Question shape | Typical stem | Usual difficulty |
|---|---|---|
| Solve | “If is the solution to the system, what is the value of ?” | Easy–medium (MCQ or SPR) |
| Combination | “What is the value of ?” or “of ?” |
3 questions on systems of two linear equations. Every option is explained once you answer.
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What does the solution to a system of two linear equations represent on a graph?
| Medium |
| Model | “Which system of equations represents this situation?” or “How many ... were sold?” | Easy–medium, in context |
| Interpret | “Which is the best interpretation of the solution ?” | Easy |
| Graph | Two lines are shown; find the solution | Easy |
| Number of solutions | “How many solutions does the system have?” or “For what value of does the system have no solution?” | Medium–hard |
In student-produced response (SPR) questions you type the answer yourself. A fraction such as -17/3 is accepted, and so is a decimal that fills the box, such as -5.667.
Each linear equation graphs as a line. The solution satisfies both equations, so it is the point where the two lines meet. Only three things can happen:
| The lines are... | Slopes | -intercepts | Number of solutions |
|---|---|---|---|
| Intersecting | Different | Anything | Exactly one |
| Parallel and distinct | Equal | Different | None |
| The same line | Equal | Equal | Infinitely many |
For and (nonzero coefficients):
In plain words: suppose one equation’s left side is a multiple of the other’s. Then the system has no solution, unless the right sides use the same multiple. In that case both equations describe the same line.
Sometimes substitution or elimination makes both variables vanish. Then read the statement that is left. An always-true statement, like , means infinitely many solutions. A false one, like , means no solution.
The SAT often asks for a combination such as , or . Before solving for each variable, try adding or subtracting the equations as given. If and , adding gives , so immediately.
Most word problems give you two totals, and each total becomes one equation.
| Context | Equation 1 | Equation 2 |
|---|---|---|
| Tickets, items, coins | Count: | Value: |
| Mixtures | Amount: | Pure part: , with rates as decimals |
| Two plans or rates | ; the solution is where the totals are equal |
-13/6 is safest. If you use a decimal, fill the space, as in -2.167.