Master two powerful methods for solving systems of linear equations: substitution and elimination, through engaging real-world scenarios.
Understanding the concept and types of solutions
A system of linear equations is a collection of two or more linear equations with the same variables that we solve simultaneously.
Example:
x + y = 5
2x - y = 1
Goal: Find values of x and y that satisfy BOTH equations
The solution is the point where the two lines intersect
One Solution: Lines intersect at one point
Example: x = 2, y = 3 (unique solution)
No Solution: Lines are parallel (never intersect)
Example: Inconsistent system
Infinitely Many Solutions: Lines are identical
Example: Dependent system
Solve one equation for one variable, then substitute into the other equation
Choose the easier equation and solve for x or y
Replace the variable in the second equation
Solve the resulting equation for the remaining variable
Use the found value to solve for the other variable
Step 1: Solve first equation for y
x + y = 5 → y = 5 - x
Step 2: Substitute into second equation
2x - (5 - x) = 1
Step 3: Solve for x
2x - 5 + x = 1 → 3x = 6 → x = 2
Step 4: Find y
y = 5 - 2 = 3
Solution: (2, 3)
Check: 2 + 3 = 5 ✓, 2(2) - 3 = 1 ✓
Add or subtract equations to eliminate one variable
Write equations in standard form: ax + by = c
Multiply equations to make one variable's coefficients equal
Add or subtract equations to eliminate one variable
Solve for one variable, then substitute to find the other
Step 1: Equations are already aligned
3x + 2y = 7
x - 2y = -1
Step 2: y coefficients are opposites (2 and -2)
No multiplication needed!
Step 3: Add equations
(3x + 2y) + (x - 2y) = 7 + (-1)
4x = 6 → x = 1.5
Step 4: Substitute to find y
1.5 - 2y = -1 → -2y = -2.5 → y = 1.25
Solution: (1.5, 1.25)
Check: 3(1.5) + 2(1.25) = 7 ✓, 1.5 - 2(1.25) = -1 ✓
Let's solve a practical problem using systems of equations
Scenario: A concert venue sells two types of tickets:
Given:
Find: How many of each type of ticket were sold?
Step 1: Define variables and write equations
Let x = VIP tickets, y = General tickets
x + y = 200 (total tickets)
50x + 25y = 7500 (total revenue)
Step 2: Solve first equation for y
y = 200 - x
Step 3: Substitute and solve
50x + 25(200 - x) = 7500
50x + 5000 - 25x = 7500
25x = 2500 → x = 100
Step 4: Find y
y = 200 - 100 = 100
Answer:
100 VIP tickets and 100 General tickets
Check: 100 + 100 = 200 ✓, 50(100) + 25(100) = 7500 ✓
Explore more complex aspects of systems of equations
One Solution: Lines intersect at one point
Example: x + y = 5, 2x - y = 1 → (2, 3)
No Solution: Parallel lines (never intersect)
Example: x + y = 3, x + y = 5 → Inconsistent system
Infinite Solutions: Same line (overlapping)
Example: x + y = 3, 2x + 2y = 6 → Dependent system
Augmented Matrix:
[1 1 | 5]
[2 -1 | 1]
Row Operations:
• Swap rows
• Multiply row by constant
• Add multiples of rows
Goal: Reduce to row-echelon form
[1 0 | 2]
[0 1 | 3] → x = 2, y = 3
Introduction to solving systems with three unknowns
Step 1: Eliminate one variable
Use two equations to eliminate x, y, or z
Step 2: Create 2×2 system
Solve the resulting two-variable system
Step 3: Back substitute
Find the third variable using original equations
System:
x + y + z = 6
2x - y + z = 3
x + 2y - z = 2
Solution:
x = 1, y = 2, z = 3
Chemistry: Balancing equations
Find coefficients for chemical reactions
Economics: Supply and demand
Market equilibrium with multiple goods
Engineering: Circuit analysis
Current flow in electrical networks
Apply both methods to solve these systems
System:
y = 2x + 1
3x + y = 10
Solution:
Substitute: 3x + (2x + 1) = 10
5x + 1 = 10 → 5x = 9 → x = 1.8
y = 2(1.8) + 1 = 4.6
Answer: (1.8, 4.6)
System:
2x + 3y = 13
2x - y = 1
Solution:
Subtract: (2x + 3y) - (2x - y) = 13 - 1
4y = 12 → y = 3
2x - 3 = 1 → 2x = 4 → x = 2
Answer: (2, 3)