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How to Solve Systems of Equations Fast on the SAT

Two equations, two unknowns — the real skill is picking the faster method in 3 seconds. Here's how to read a system and choose substitution or elimination instantly, plus the Desmos shortcut: the solution is where the lines cross.

Math · Systems of two linear equations2:10Published July 5, 2026

On YouTube: SAT Systems: Stop Using the Slow Method

What this lesson covers

A variable already isolated? Substitute.
Coefficients match? Eliminate (add or subtract).
Desmos: graph both lines, the solution is the intersection
Practice: x + y = 7, 2x − y = 5
The three traps (solving for only one variable, add vs subtract, reusing the same equation)

Questions worked in the video

0:29y = 2x − 1 and 3x + y = 14 — solve the system (substitution).
0:292x + 3y = 12 and 2x − y = 4 — solve the system (elimination).
1:09x + y = 7 and 2x − y = 5 — solve the system.

Worked examples

The questions the video works, written out: the setup, each step, the answer and the trap.

0:29Example 1

Solve the system $y = 2x - 1$ and $3x + y = 14$ .

Read the system before touching it. The first equation already has y alone on one side, so substitution is the fast method.
Replace y in the second equation with 2x - 1: $3x + (2x - 1) = 14$ .
Combine: $5x - 1 = 14$ , so $5x = 15$ and x = 3.
Find y from the equation you have not used yet: $y = 2(3) - 1 = 5$ .
Check in the other equation: $3(3) + 5 = 14$ . In Desmos, graphing both lines and clicking where they cross reads (3, 5).

Answer: (3, 5), that is x = 3 and y = 5

An isolated variable is a ready-made substitution: dropping 2x - 1 in for y turns two equations into one equation in one unknown. The trap is stopping at x = 3; the SAT usually asks for y, or for x + y, so a system is not solved until both values are found. A second trap is substituting back into the same equation you just used, which only returns 0 = 0.

0:29Example 2

Solve the system $2x + 3y = 12$ and $2x - y = 4$ .

Both equations have 2x with the same sign, so elimination is faster than isolating a variable: subtract the second equation from the first.
$(2x + 3y) - (2x - y) = 12 - 4$ . The 2x terms cancel, and $3y - (-y) = 4y$ , so $4y = 8$ and y = 2.
Put y = 2 into either equation: $2x - 2 = 4$ , so $2x = 6$ and x = 3.
Check in the first equation: $2(3) + 3(2) = 6 + 6 = 12$ .

Answer: (3, 2), that is x = 3 and y = 2

Matching coefficients cancel when you subtract, so one line of arithmetic removes x entirely. The trap is adding when you should subtract: adding these gives 4x + 2y = 16, which still has both unknowns and gets you nowhere. Also watch the double negative, since subtracting -y adds a y, giving 4y and not 2y.

1:09Example 3

Solve the system $x + y = 7$ and $2x - y = 5$ .

The y terms are +y and -y, opposite coefficients, so adding the equations eliminates y at once.
$(x + y) + (2x - y) = 7 + 5$ , so $3x = 12$ and x = 4.
Back into the first equation: $4 + y = 7$ , so y = 3.
Check in the second: $2(4) - 3 = 8 - 3 = 5$ .

Answer: (4, 3), that is x = 4 and y = 3

Opposite coefficients mean add, matching coefficients mean subtract; here +y and -y cancel by addition, so nothing needs multiplying first. The trap is subtracting out of habit, which gives -x + 2y = 2 and leaves both unknowns in play. Once x = 4 is known, y comes from the simpler equation in one step.

Chapters

Lesson transcript

The narration of the video, word for word, under its chapter headings.

0:00Pick the fast method

Welcome to SAT Climb. Two equations, two unknowns. The trick isn't just solving — it's picking the faster method in 3 seconds. Substitution, or elimination?

0:16The setup: two systems

Two systems. For each, one method is faster. The clue is how the equations are already written.

0:29Substitution vs elimination

If a variable is alone — like y = 2x − 1 — substitute it in. 3x + that y is 14, so x = 3, y = 5. If coefficients match — 2x in both — eliminate: subtract, and x is gone. y = 2, x = 3.

0:49The Desmos intersection

Or graph both lines. The solution is the one point where they cross. Type both equations, click the intersection, and Desmos gives it exactly: (3, 5).

1:09Your turn

Your turn. x + y = 7; 2x − y = 5. The y's cancel — add the equations. What's the solution?

1:35Three traps

Three traps. Solving for x but forgetting y — a system needs both. Adding when you should subtract, or the reverse. And substituting back into the same equation, which tells you nothing.

1:56Recap

Systems: solved. Pick the method, and it falls apart in seconds. Start free at satclimb.com.

Read the written version: the Systems of two linear equations strategy guide, then try 25 hard Systems of two linear equations questions with full explanations. Both are free, no account needed.

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