Systems of Equations: Substitution and Elimination

A system of equations is two or more equations with the same variables. The solution is the (x, y) pair that satisfies both. For example, the lines y = 2x + 1 and y = −x + 4 intersect at one point — that point is the solution.

Method 1: Substitution

Solve one equation for one variable, then substitute that expression into the other equation.

Example:

Equation 1 already gives y in terms of x. Substitute into equation 2:

Now plug x = 1.5 into equation 1 to find y:

Solution: (1.5, 4)

Check: 2(1.5) + 4 = 3 + 4 = 7 ✓

When one variable is easy to isolate, substitution works well.

Method 2: Elimination (Addition)

Add or subtract the equations so one variable cancels out. You may need to multiply one or both equations first.

Example:

Subtract equation (2) from equation (1) to eliminate x:

Plug y = 2 into (2): 2x − 2 = 4 → 2x = 6 → x = 3

Solution: (3, 2)

Example when you need to multiply first:

To eliminate x: multiply (1) by 2 and (2) by −3, then add.

Plug y = 2 into (1): 3x + 4 = 16 → 3x = 12 → x = 4

Solution: (4, 2)

Which Method to Use?

• Substitution — when one equation is already solved for a variable (e.g. y = 2x + 1)
• Elimination — when coefficients line up nicely or you can easily make them match

Special Cases

• No solution — parallel lines (same slope, different y-intercept)
• Infinitely many solutions — same line (equivalent equations)

Common mistakes

Dropping a negative: When multiplying an equation by a negative to eliminate a variable, every term must be multiplied — including the constant on the right side.

Forgetting to solve for both variables: After finding one variable, substitute back to find the other. The answer is a pair (x, y).

More examples

Example (substitution): y = 2x + 1 and 3x + y = 11

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