Solving Systems by Substitution
Key steps, strategies, and PSAT tips for solving systems of equations by substitution.
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Questions Covered in This Set
10 cards to master
What is a system of equations?
Two or more equations sharing the same variables; the solution is the pair (x, y) that makes all equations true — graphically, the intersection point.
What are the four steps of substitution?
1) Isolate a variable, 2) Substitute that expression into the OTHER equation (use parentheses), 3) Solve the one-variable equation, 4) Back-substitute and state the ordered pair.
Which variable should you isolate first?
One with coefficient 1 or -1, so isolating it creates no fractions.
Solve: y = 2x + 1 and 3x + y = 11
3x + (2x+1) = 11 → 5x = 10 → x = 2, y = 5. Solution (2, 5).
Solve: x - 2y = 4 and 3x + y = 5
x = 2y + 4 → 3(2y+4) + y = 5 → 7y = -7 → y = -1, x = 2. Solution (2, -1).
When is elimination better than substitution?
When every coefficient is large/ugly (no 1 or -1), since substitution would create fractions.
Most common PSAT mistake on systems?
Solving correctly but answering the wrong quantity — they may ask for x + y or 2y, not just x.
Solve the non-linear system: y = x² - 3 and y = 2x
2x = x² - 3 → x² - 2x - 3 = 0 → (x-3)(x+1)=0 → x = 3 or -1; points (3, 6) and (-1, -2).
How do you check a substitution answer quickly?
Plug the ordered pair into the equation you did NOT use for substituting.
If y = 5 - x and 3x - y = 7, what is x + 2y?
x = 3, y = 2, so x + 2y = 7.
