1. 3.2a – Solving Systems algebraically

2. To solve systems using Substitution:
Step #1: Check to see if one of the equations has a variable alone
Step #2: Substitute what that variable equals into the other equation
Step #3: Solve for the other variable
Step #4: When you know what one variable is, plug it back in to find the other one.
Step #5: Write the answer as an ordered pair (x, y)

3. y = -2x + 1 – x + 2(-2x + 1) = 2 – x – 4x + 2 = 2 y = -2(0) + 1
Ex#1 Solve the system by substitution. Check by graphing
y = -2x + 1
– x + 2(-2x + 1) = 2
– x – 4x + 2 = 2
y = -2(0) + 1
-5x + 2 = 2
y = 0 + 1
y = 1
-5x = 0
(x, y)
x = 0
(0, 1)

4. (0, 1)

5. Ex#2 Solve the system by substitution.
y = -2x + 13
4x – 3(-2x + 13) = 11
4x + 6x – 39 = 11
y = -2(5) + 13
10x – 39 = 11
y =
y = 3
10x = 50
(x, y)
(5, 3)
x = 5

6. Try These: Solve the system by substitution.
x = 2y – 7
3(2y – 7) + 4y = 9
x = 2(3) – 7
6y – y = 9
x = 6 – 7
10y – 21 = 9
x = -1
10y = 30
(x, y)
(-1, 3)
y = 3

7. Ex#2 Solve the system by substitution.
6x – 3(2x – 5) = 15
6x – 6x + 15 = 15
15 = 15
Infinite solutions

8. Ex#2 Solve the system by substitution.
6x + 2(-3x + 1) = 5
6x – 6x + 2 = 5
2  5
No Solution

9. To eliminate a variable line up the variables with x first, then y
To eliminate a variable line up the variables with x first, then y. Afterward make one of the variables opposite the other. You might have to multiply one or both equations to do this.

10. 2 3x – y = 8 6x – 2y = 16 x + 2y = 5 x + 2y = 5 7x + 0 = 21 7x = 21
Example #3: Find the solution to the system using elimination. 2
3x – y = 8
6x – 2y = 16
x + 2y = 5
x + 2y = 5
7x + 0 = 21
7x = 21
7 7
3 + 2y = 5
x = 3
2y = 2
y = 1
(x, y)
(3, 1)

11. -3 2x + y = 6 -6x – 3y = –18 4x + 3y = 24 4x + 3y = 24 –2x = 6 x = – 3
Example #3: Find the solution to the system using elimination. -3
2x + y = 6
-6x – 3y = –18
4x + 3y = 24
4x + 3y = 24
–2x = 6
x = – 3
2(–3) + y = 6
-6 + y = 6
y = 12
(x, y)
(–3, 12)

12. Example #3: Find the solution to the system using elimination.
3x – 4y = 16
9x – 12y = 48 2
5x + 6y = 14
10x + 12y = 28
19x = 76
x = 4
5(4) + 6y = 14
20 + 6y = 14
6y = -6
(x, y)
y = -1
(4, -1)

13. 5 -3x + 2y = -10 -15x + 10y = -50 3 5x + 3y = 4 15x + 9y = 12
Example #3: Find the solution to the system using elimination. 5
-3x + 2y = -10
-15x + 10y = -50 3
5x + 3y = 4
15x + 9y = 12
19y = -38
y = -2
5x + 3(-2) = 4
5x – 6 = 4
5x = 10
x = 2
(x, y)
(2, -2)

14. 12x – 3y = -9 12x – 3y = -9 3 -4x + y = 3 -12x + 3y = 9 0 = 0
Example #3: Find the solution to the system using elimination.
12x – 3y = -9
12x – 3y = -9 3
-4x + y = 3
-12x + 3y = 9
0 = 0
Infinite solutions

15. 6x + 15y = -12 6x + 15y = -12 3 -2x – 5y = 9 -6x – 15y = 27 0 = 15
Example #3: Find the solution to the system using elimination.
6x + 15y = -12
6x + 15y = -12 3
-2x – 5y = 9
-6x – 15y = 27
0 = 15
No solution