Section 1.3: Systems of Equations
This section covers solving systems of linear equations using substitution, elimination, and graphing methods. Applications in word problems are included.
Example 1: Solving by Substitution
Solve the system: \( x + y = 5 \), \( 2x - y = 1 \).
Step 1: Solve first equation for y: \( y = 5 - x \)
Step 2: Substitute into second: \( 2x - (5 - x) = 1 \) → \( 3x - 5 = 1 \)
Step 3: Solve: \( 3x = 6 → x = 2 \)
Step 4: Find y: \( y = 5 - 2 = 3 \)
Example 2: Solving by Elimination
Solve the system: \( 3x + 2y = 12 \), \( 2x - 2y = 2 \).
Step 1: Add the equations: \( 3x + 2y + 2x - 2y = 12 + 2 → 5x = 14 → x = 14/5 \)
Step 2: Substitute x into first equation: \( 3(14/5) + 2y = 12 → 42/5 + 2y = 12 → 2y = 18/5 → y = 9/5 \)
Practice Problems
- Solve \( x + 2y = 8 \), \( 3x - y = 5 \)
- Solve \( 2x + 3y = 7 \), \( 4x - y = 5 \)
- Solve \( x - y = 4 \), \( 2x + y = 10 \)
- Word problem: A theater sold 100 tickets. Adult $10, student $6. Total $760. Find numbers sold.
- Word problem: Mix 30 liters of solution, one 5$/L, other 8$/L. Cost 6.50$/L. How much of each?