Solving Systems of Linear Equations

Solving Systems of Equations
by Graphing

Example: Solve the system of equations by graphing:
x - 2y = 1
x + y = 4
Solve each for y:

y = ½x - ½
y = -x + 4

Solving Systems of Equations
Algebraically

Example: Solve by substitution:

3x + 6 = 7y
x + 2y = 11

Solve one for either variable:
x = 11-2y then substitute (11-2y) in for x in the other equation:

3(11-2y) + 6 = 7y
33 - 6y + 6 = 7y
39 = 13y
3 = y

x = 11-2(3)
x = 5

Point of intersection is (5,3)

Example: Solve by elimination:

x- 2y = 0
y = 2x - 3

Arrange the equation in standard form:
x-2y = 0
-2x + y = -3

Multiply bottom equation by 2 to cancel out ys:

x - 2y = 0
-4x + 2y = -6 
then combine like terms to find x
-3x = -6
x = 2

2 - 2y = 0
-2y = -2
y = 1

the point of intersection is (2,1)

Application of Systems of Equations:

If the Astros sold 15 hats and 12 shirts the first hour of the game and made $364.50. Then sold 10 hats and 9 shirts and made $255.25. How much was each hat and each shirt?

x = hats and y = shirts

15x + 12y = 364.50
10x + 9y = 255.25

Use elimination by multiplying top equation by -3 and bottom by 4
-45x - 36y = -1093.5
40x +36y = 1021

the y coordinate cancels then combine the x and constant

-5x = -72.5
x = 14.50

plug x back in to either equation to get y

10(14.5) + 9y = 255.25
145+ 9y = 255.25
9y = 110.25
y = 12.25

therefore hats costs $14.50 and shirts costs $12.25

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Graphing Systems of Inequalities

y > x - 2
y < -2x + 4

Graph each and shade appropriate area:

Example Problems

1. Solve the system by graphing: x + y = 5
                                                    2x - y = 4

2. Solve the system using substitution:    5x + 3y = -4
                                                                  7x - y = 36

3. Solve the system using elimination:    2x - y = 4
                                                                2x + 3y = 12

4. Solve the system using either method: x + y = 6
                                                                3x - 4y  = 4

5. Graph the inequality:    y ³ -x + 5
                                        y> 1½x - 5