Systems of linear equations
Two equations, two unknowns, one point where it all comes together. Master substitution, elimination and the no-solution tricks that show up on every SAT.
The solution is where the lines meet
A system is two equations that must both be true at once. Each equation is a line, so the solution is the point on both lines: where they cross. Two lines can meet once, never (parallel), or everywhere (the same line).
Substitution
If one equation already tells you what y (or x) equals, swap that expression into the other equation. You are left with one equation in one variable, which you already know how to solve.
- Replace y in the second equation: 3x + 2(x + 4) = 23
- Expand and combine: 5x + 8 = 23, so x = 3
- Go back for y: y = 3 + 4 = 7. The solution is (3, 7).
- Check in the second equation: 3(3) + 2(7) = 9 + 14 = 23.
Elimination
When both equations are in Ax + By = C form, add or subtract the equations so one variable disappears. If the coefficients do not match yet, multiply one equation first.
- The y terms are opposites (+3y and −3y). Add the equations: 6x = 18
- x = 3. Substitute: 2(3) + 3y = 12, so 3y = 6
- y = 2. Solution (3, 2)
- Multiply the second equation by 2: 6x − 2y = 14
- Add to the first: 7x = 21, so x = 3
- Substitute: 3 + 2y = 7, so y = 2. Solution (3, 2)
Choosing the fastest method
Both methods always work, but one is usually quicker. Substitution when a variable is already isolated. Elimination when coefficients match or are easy to match. Desmos when the numbers are messy.
One, none or infinitely many
You can count solutions without solving. Rewrite both as y = mx + b, or compare coefficients:
- Different slopes: the lines cross once. Exactly one solution.
- Same slope, different intercepts: parallel lines. No solution.
- Same slope and same intercept: the same line. Infinitely many solutions. (One equation is a multiple of the other.)
Systems in real life
Word problems with two unknowns usually give you two facts: one about how many and one about how much. Turn each fact into an equation, then solve the system.
Answer what is asked
SAT systems questions often ask for an expression such as x + y or 2x − y. Sometimes adding or subtracting the whole equations gives it to you directly. And always double-check you are giving the variable they asked for.
Spot the trap
Each solution below has one faulty step. These are the exact slips SAT wrong answers are built from.
Desmos moves
Systems are where Desmos shines. Graph both equations and tap the intersection. It works in any form, and with a slider it can even find missing constants.