1. 26. Solving Matrix Equations

2. Solving Matrix Equations
Suppose Ax = B
How do you solve for x?
We cannot divide
matrices, but we
can multiply by the inverse.
Note: Inverse must be on left side.
X = A-1B

3. Solving Matrix Equations
Step 1: Find A-1
Step 2: Multiply both sides of the equation by A-1 on the left

4. Solving a Matrix Equation

5. Step 2: Multiply both sides of the equation
Step 1: Find A-1
A-1=
Step 2: Multiply both sides of the equation
By A-1 on the left

6. Solution by multiplying
You can check the
Solution by multiplying
AX to see if you get B

7. Ex. You try!! Solve

8. Example
Use an inverse matrix to solve this system.
x-2y=0
2x-3y = 3

9. Matrix Equations
Example: Solve the system 3x - 2y = x + 2y = -5

10. Matrix Equations
Example #2: Solve the 3 x 3 system 3x - 2y + z = 9 x + 2y - 2z = -5 x + y - 4z = -2
Using a graphing calculator,

11. Matrix Equations
Example #2, continued

12. Word Problem Example 1
Kevin would like to buy 10 bouquets. The standard bouquet costs $7, and the deluxe bouquet costs $12. He can only afford to spend $100. How many of each type can he buy?
Define Variables:
X: standard bouquet Y: deluxe bouquet
Equation 1 Cost: x+12y=100
Equation 2 Bouquets :
x+y=10
Solution: (4,6)

13. Word Problem Example 2
A hot air balloon is 10 meters above the ground and rising at a rate of 15 meters per minute. Another balloon is 150 meters above the ground and descending at a rate of 20 meters per minute. When will the two balloons meet?
Define Variables:
x=minutes y=height in meters
Equation 1
y=15x+10
Equation 2
y=-20x+150
Solution: (4,70)

14. Word problem Example 3
A group of 3 adults and 10 students paid $102 for a cavern tour. Another group of 3 adults and 7 students paid $84 for the tour. Find the admission price for an adult ticket and a student ticket.
Define Variables:
x= adult ticket price y=student ticket price
Equation 1
3x+10y=102
Equation 2
3x+7y=84
Solution (14,6)

15. An Algebra Test contains 38 problems
An Algebra Test contains 38 problems. Some of the problems are worth 2 points each. The rest of the questions are worth 3 points each. A perfect score is 100 points. How many problems are worth 2 points? How many problems are worth 3 points?
Define Variables:
x=2 pt. questions y=3 pt. questions
Equation 1
x+y=38
Equation 2
2x+3y=100
Solution (14,24)

16. Example
The sum of the ages of Ryan and his father is 66. His father is 10 years more than 3 times as old as Ryan. How old are Ryan and his father?
Define Variables
x=Ryan’s age y=Dad’s age
Equation 1
x+y=66
Equation 2
y=3x+10
Solution (14,52)

17. Assessment
Does taking variables out of problems make them easier or more difficult?