Step by Step
S
Substitution — solve one variable, plug it in
Solve one equation for one variable, then substitute that expression into the other equation, leaving only one variable to solve for.
1
When substitution works best
Substitution is usually fastest when one equation is already solved for a variable, or can be easily rearranged that way.
E
Elimination — add or subtract equations
Add or subtract the two equations (after possibly multiplying one or both by a constant) so that one variable cancels out entirely.
2
When elimination works best
Elimination is usually fastest when the coefficients of one variable are already equal or easy to make equal by multiplying.
Applied Walkthrough
1
Solve the system: y = 2x + 1 and 3x + y = 11, using substitution since y is already isolated.
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Substitute y = 2x + 1 into the second equation: 3x + (2x + 1) = 11, which simplifies to 5x + 1 = 11, so x = 2.
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Plug x = 2 back into y = 2x + 1: y = 5. The solution is (2, 5).
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If instead both equations were in standard form (like 3x + 2y = 12 and 3x − y = 3), elimination would be faster — subtracting the equations would immediately cancel the x terms.
Exam Application
Exams test whether you can choose the more efficient method (substitution vs. elimination) based on how the system is set up, and whether you can correctly execute either method to find both variable values.
⚠ Common Trap
A common trap is defaulting to one method regardless of the system's setup, when the other method would be much faster. Look at the equations first — if one variable is already isolated, substitution is faster; if coefficients line up easily, elimination is faster.
✓ Quick Self-Check
1. What are the two methods for solving a system of equations covered here?
Substitution and elimination.
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2. When is substitution usually the faster method?
When one equation is already solved for a variable, or can easily be rearranged that way.
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3. When is elimination usually the faster method?
When the coefficients of one variable are already equal or easy to make equal.
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4. Solve y = 2x + 1 and 3x + y = 11 using substitution.
x = 2, y = 5.
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5. What do you do to a variable's coefficient before adding/subtracting equations in elimination, if needed?
Multiply one or both equations by a constant so that variable's coefficients match (or are opposites).
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