3.2 Systems of Linear Equations

1. Learning Objectives

Define systems of linear equations in two variables.
Solve systems using Substitution, Elimination, and Graphical methods.
Identify unique, infinite, or no solution systems.

A system of linear equations consists of two equations with variables x and y.

Condition	Graph Shape	Number of Solutions
Slopes are different	Intersecting Lines	1 Unique Solution (x, y)
Slopes equal, Intercepts different	Parallel Lines	No Solution
Slopes equal, Intercepts equal	Coinciding Lines	Infinitely Many Solutions

Example 1 (Elimination): Solve $x + y = 10 x - y = 4$

Explanation: Add the equations to cancel y.
2x = 14 → x = 7.
Substitute x=7 into the first equation: 7 + y = 10 → y = 3.
Result: (7, 3)

1. Solve: x + y = 10 and x - y = 4.

Add equations: 2x = 14 → x = 7. Sub x into first: 7 + y = 10 → y = 3. Ans: (7, 3)

2. Solve using substitution: y = 3x and x + y = 12.

Sub 3x for y: x + 3x = 12 → 4x = 12 → x = 3. Then y = 3(3) = 9. Ans: (3, 9)

3. How many solutions if lines have the same slope but different intercepts?

Same slope means parallel lines. Parallel lines never intersect. Ans: No Solution.

4. Solve: 2x + y = 8 and y = 2.

Sub y=2: 2x + 2 = 8 → 2x = 6 → x = 3. Ans: (3, 2)

5. Solve: x = y and x + y = 10.

Sub y for x: y + y = 10 → 2y = 10 → y=5. Since x=y, x=5. Ans: (5, 5)

6. Solve by elimination: 2x - y = 5 and x + y = 4.

Add eq: 3x = 9 → x = 3. Sub x into second: 3 + y = 4 → y = 1. Ans: (3, 1)

7. Solve: y = 2x - 1 and y = -x + 5.

Set equal: 2x - 1 = -x + 5 → 3x = 6 → x=2. y = 2(2)-1 = 3. Ans: (2, 3)

8. If equations are 2x + y = 4 and 4x + 2y = 8, what type of system is it?

Eq2 is Eq1 × 2. They represent the same line. Ans: Infinitely Many Solutions.

9. Intersection of x = 4 and y = -3?

Vertical and horizontal lines meet at the point defined by their constants. Ans: (4, -3)

10. Sum is 20, difference is 10. Find numbers.

x+y=20, x-y=10. Add: 2x=30 → x=15. Sub: 15+y=20 → y=5. Ans: (15, 5)

11. Solve: 3x + y = 10 and x = 2.

Sub x=2: 3(2) + y = 10 → 6 + y = 10 → y = 4. Ans: (2, 4)

12. Solve: x - y = 0 and x + y = 0.

Add eq: 2x = 0 → x=0. Then 0+y=0 → y=0. Ans: (0, 0)

13. Solve: 4x + 2y = 12 and y = -2x + 6.

Eq2 rearranged is 2x+y=6. Eq1 is 2(2x+y)=12. Same line. Ans: Infinite Solutions.

14. Sum of ages of A and B is 30. A is twice B.

a+b=30, a=2b. Sub: 2b+b=30 → 3b=30 → b=10, a=20. Ans: (20, 10)

15. Solve: x + 2y = 8 and x = 0.

Sub x=0: 0 + 2y = 8 → 2y = 8 → y = 4. Ans: (0, 4)

16. Solve: 2x - y = 3 and 2x - y = 7.

LHS is same, RHS is different. Lines are parallel. Ans: No Solution.

18. Solve: x = 2y and x + y = 6.

Sub 2y for x: 2y + y = 6 → 3y=6 → y=2, x=4. Ans: (4, 2)

19. One number is 3 more than another. Sum is 11.

x = y + 3, x + y = 11. Sub: (y+3)+y=11 → 2y=8 → y=4, x=7. Ans: (7, 4)