Systems of Two Linear Equations

Unit 1 · Algebra

What the SAT asks here

A system of two linear equations asks for the (x, y) pair that satisfies both. The SAT expects you to choose between substitution and elimination based on which is faster, and to recognise the three solution scenarios (one, none, infinite). Expect 2–3 problems per test.

Three solution scenarios

Lines intersect → exactly one solution

Same slope, different intercepts → no solution

Same slope and intercept → infinitely many solutions

Two algebraic methods

Substitution — solve one equation for a variable, substitute

Elimination — add/subtract multiples to cancel a variable

SAT Tip: Pick elimination when one variable's coefficients are easy to match (or already are). Pick substitution when one equation already isolates a variable.

Caution: The Digital SAT lets you graph systems on the built-in Desmos calculator — use it as a sanity check, but practice the algebra so you're not slow when an algebraic question rules graphing out.

Type 1

Solve a system

Apply substitution or elimination to find the unique (x, y). Add or subtract the equations so one variable cancels, then back-substitute.

Example 1

Solve the system. 2x + y = 7 and x − y = −1.

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Type 2

Number of solutions

Rewrite both equations in slope-intercept form. Compare slopes (and intercepts when slopes match) to count solutions.

Example 2

For what value of c does the system 6x + 2y = 10 and 3x + cy = 8 have no solution?

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Type 3

Set up from context

Translate a word problem into two simultaneous equations. Define your variables explicitly first, then write both constraints.

Example 3

A coffee shop sells small drinks for $3 and large drinks for $5. One morning they sold 80 drinks for $320 total. How many large drinks were sold?

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