1. Chapter 5: Systems
Vogler
Algebra II

2. Solving Systems: Graphing
Solutions to all systems are found where the lines cross.
In this case, the solution is where x=8, y=2.4
(8, 2.4)

3. Solving Systems: Substitution
Since the solution to a system is where the lines cross, then we can also solve systems by solving both equations for where they are equal.
This works best when we have at least one equation equal to y or x.

4. Solving systems: Substitution
Joey walks at 2MPH. Sally walks at 1 MPH. Joey started 3 miles from home, Sally started 5 miles from home. At what time and distance from home do they meet?

5. Solving Systems: Substitution
Make the equations equal to each other and solve:
2x+3=x+5
-x+2x+3=x+5-x
X+3=5
X+3-3=5-3
X=2
They are also 7 miles from home (2+5=7)

6. Solving Systems: Substitution
Susan makes $12 each hour. Sam is paid $2500 in salary. After how many hours do they make the same amount? How much do they make at this point?

7. Solving Systems: Combinations
Add (combine) the equations to get rid of a variable:
12x+4y=24
5x+2y=14
-2(5x+2y=14)
12x+4y=24
-10x-4y=-28
2x = -4
x = -2
5•-2+2y=14
-10+2y=14
2y=24
y=12
(-2, 12)

8. Solving Systems: Combinations
To attend a movie, each child is charged $4 and each adult is charge $7. The night’s revenue was $ If 500 people attended movies that evening, how many children and adults attended?

9. Solving Systems: Matrices
To solve a system using a matrix, find the inverse of the system:
2x+3y=7
3x-y=5
[2 3]-1=
[3 -1]
Then multiply:
(2, 1)

10. Solving Systems: Matrices
Solving systems with matrices will be much easier with a graphing calculator
For all math problems in this class, you may use whatever method you wish to solve

11. Graphing Inequalities: Linear inequalities
Graph the line
Y<-3x+15
If y<… then shade below
If y>… then shade above
Check using the origin (0,0)
0<-3(0)+15
0<15, we shaded correctly

12. Graphing Inequalities: Systems
6x+10y<60
Y<2x
The feasible
region is
where the two
shaded areas
overlap

13. Graphing Inequalities: Systems
Widgets for midgets sell lifters for $100 and extendees for $7. However, they only have $1500 to spend. If lifters are $50 each and extendees are $5 each. There are only 500 hours available to make each. Lifters require 20 hours to make and extendees require 5 hours. How many of each should be made to maximize the profit?

14. Graphing Inequalities

15. Graphing Inequalities: Linear Programming
From the last example, find the point that uses up all $1500 in the most equitable way possible (meaning that more than one extendee and more then one lifter must be made).
Use the end points and intersection points of your graph to solve the problem

16. Graphing Inequalities: Linear Programming
Jovan can make 10 cakes in an hour or he can make 20 muffins in an hour. Each cake can be sold for $5 and each muffin can be sold for $2. If he has 10 hours and $20, and each cake costs $3 and each muffin costs $1 to make, what combination of muffins and cakes should he make?