This document provides 3 examples of solving systems of linear equations using different methods: elimination, substitution, and graphing. The first example has a unique solution of x=3 and y=7. The second example has no solution. The third example has infinitely many solutions since the lines are coincident.

1. Examples

2. Example 1
5x+4y=43
2x+y=13
• Solution=3,y=7

3. Elimination Method
2(5x+4y)=43=10x+8y=86
5(2x+y)=13=10x+5y=65
(10x+8y)-(10x+5y)=3y=21
Y=21/3=7
13-(7)=6
X=6/2=3

4. Substitution Method
Y=13-2x
5x+4(13-2x)=43
5x+52-8x=43
-3x+52=43
3x=52-43=9
X=9/3=3
Y=13-2(3)=7

5. Graph Method
X 1 2 3 4 5
y 9.5 8.25 7 5.75 4.5
5x+4y=43
2x+y=13
X 1 2 3 4 5
y 11 9 7 5 3

6. Example 2
5x-5y=7
6x-6y=12
• No Solution

7. Elimination Method
6(5x-5y)=30x-30y=42
5(6x-6y)=30x-30y=60
(30x-30y)-(30x-30y)=0=18

8. Substitution Method
X=7+5y/5
6(7+5y/5)-6y=12
42+30y-30y=60
42=60

9. Graph Method
5x-5y=7
6x-6y=12
X 1 2 3 4 5
y -0.4 0.6 1.6 2.6 3.6
X 1 2 3 4 5
y -1 0 1 2 3

10. Example 3
4x+8y=24
7x+14y=42
• Infinitely Many Solutions

11. Elimination Method
7(4x+8y)-4(7x+14y)=28x+56y-28x-56y=0

12. Substitution Method
X=24-8y/4
7(24-8y/4)+14y=42
168-56y+56y=168
168=168

13. Graph Method
• 4x+8y=42
• 7x+14y=42
X 1 2 3 4 5
y 4.75 4.25 3.75 3.25 2.75
X 1 2 3 4 5
y 4.75 4.25 3.75 3.25 2.75