1. Systems of Equations and Inequalities
Graphing Systems of Equations
Solving Systems by Substitutions
Solving Systems by Eliminations
Solving Systems by Elimination using multiplication
Applying Systems of Linear Equations
Systems of Inequalities

2. Graphing Systems of Equations
The goal is to find a value for x and y that make two equations true. Example
y = 3x - 1
x + y = 7
step 1 Graph each equation
step 2 Notice the point of intersection
step 3 Find the coordinates of the point of intersection.
What do these coordinates mean?

3. Solve Systems by Substitution
Think about parallel lines. If your two equations are parallel lines you will either have infinitely many solutions or no solutions.
Example: The sum of the measures of angles X and Y is 180 (They are supplementary). The measure of angle X is 24 greater than angle Y.
Equation one. “The sum of X & Y is 180”: x + y = 180
Equation two. “Angle X is 24 greater than y”: x = 24 + y
Since we have x equal to something, we can substitute what x is equal to in the first equation.
(24 + y) + y = 180: notice how x is now 24 + y
24 + 2y = Solve for y: 156 = 2y; y = 78.
Now we can substitute 88 for y in either x = 24 + y or x + y = 180
Either we get x = ; x=102 Or we get x + 78 = x = 102

4. Elimination using Addition and Subtraction
You can add or subtract equations with each other.
x + y = 4
x – y = 8
2x = 12: x = 6 When we substitute 6 for y what does y equal?
Subtract
x + 4y = -4
- x + 10y = -16
-6y = 12: y = -2 Notice 4 – 10 = -6 and - 4 – (-16) = 12

5. Elimination using Multiplication
Prior to adding equations together, you can multiply an entire equation by a constant.
x + y = 2 We will take this equation and multiply by 3
-3x + 4y = 15
3x +3y = 6
7y = 21
There are instances where you may need to multiply 2 equations by a constant.
8x + 3y = -7
7x + 2y = -3

6. Applying Systems of Linear Equations
Manuela has a total of 40 DVDs of movies and television shows. The number of movies is 4 less than 3 times the number of television shows. Write and solve a system of equations to find the numbers of movies and television shows that she has on DVD.

7. Another word problem
An online music club offers individual songs for one price or entire albums for another. Kendrick pays $14.90 to download 5 songs and 1 album. Geoffrey pays $21.75 to download 3 songs and 2 albums.
How much does each song cost and each album?

8. Systems of Inequalities
Graphing an inequality.
Exactly like graphing an equation except there is an inequality sign.
After graphing the linear equation, decide:
1: If the line is dotted or solid
2: Which side of the line to shade
If the point (0,0) is not on the line
try it in the inequality.
0 > 0 – 2: 0 > -2 is true, shade that side
0 ≤ 2(0)+2: 0 ≤ 2 is true, shade that side

9. Solution sets of systems of inequalities
Graph each inequality with the proper shading.
Look to see where there is an intersection of regions.
That intersection is the solution region.

10. Try these systems
Y > -x x – y ≥ 3
Y < 2x x – 4y < 1