3.2a – Solving Systems algebraically

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)

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 -2 -2 y = 1 -5x = 0 -5 -5 (x, y) x = 0 (0, 1)

(0, 1)

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 = -10 + 13 +39 +39 y = 3 10x = 50 10 10 (x, y) (5, 3) x = 5

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

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

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

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.

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 -3 -3 2y = 2 y = 1 (x, y) (3, 1)

-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)

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 -20 -20 6y = -6 (x, y) y = -1 (4, -1)

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)

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

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