Section 5.2: Solving Systems by Elimination
Elimination is a method to solve a system of linear equations by adding or subtracting equations to eliminate one variable, allowing you to solve for the other variable.
Example 1
Solve the system:
\( x + y = 7 \)
\( 2x - y = 4 \)
Step 1: Add the two equations to eliminate \( y \): \( (x + y) + (2x - y) = 7 + 4 \)
Step 2: Simplify: \( 3x = 11 \) → \( x = \frac{11}{3} \)
Step 3: Substitute back into \( x + y = 7 \): \( \frac{11}{3} + y = 7 \) → \( y = 7 - \frac{11}{3} = \frac{10}{3} \)
Solution: \( (x,y) = \left(\frac{11}{3}, \frac{10}{3}\right) \)
Example 2
Solve:
\( 3x + 2y = 16 \)
\( 5x - 2y = 4 \)
Step 1: Add the two equations to eliminate \( y \): \( (3x + 2y) + (5x - 2y) = 16 + 4 \)
Step 2: Simplify: \( 8x = 20 \) → \( x = \frac{20}{8} = \frac{5}{2} \)
Step 3: Substitute back into \( 3x + 2y = 16 \): \( 3(\frac{5}{2}) + 2y = 16 \) → \( \frac{15}{2} + 2y = 16 \) → \( 2y = \frac{17}{2} \) → \( y = \frac{17}{4} \)
Solution: \( (x,y) = \left(\frac{5}{2}, \frac{17}{4}\right) \)
Practice Problems
- Solve \( 2x + y = 9 \) and \( 3x - y = 4 \).
- Solve \( 4x + 5y = 20 \) and \( 2x - 3y = -4 \).
- Solve \( x - y = 6 \) and \( 2x + y = 10 \).
- Solve \( 5x + 2y = 18 \) and \( 3x - 2y = 6 \).
- Solve \( x + 4y = 12 \) and \( 2x - y = 3 \).