1. Systems of Equations and Inequalities
C-5
Solving Systems of Equations in Three Variables

2. ACT WARM-UP Which of the following is equivalent to (3xa)b?
A) 3xab B) 3xa+b C) 3bxab D) 3bxab E) 3bxa+b
Power of a Power rule states that each base is raised to an exponent of b, therefore multiply each exponent by a factor of b. The answer is D) 3bxab.

3. Objective
Solve systems of linear equations in three variables

4. Essential Question
How do you solve a system of equations in three variables using a calculator?

5. A Global Positioning System (GPS) gives locations using the three coordinates of latitude, longitude, and elevation. You an represent any location in three-dimensional space using a three-dimensional coordinate system, sometimes called coordinate space. Each point in coordinate space can be represented by an ordered triple of the form (x, y, z). The system is similar to the coordinate plane but has an additional coordinate based on the z-axis.

6. Recall that the graph of a linear equation in two dimensions is a straight line. In three-dimensional space, the graph of a linear equation is a plane. When you graph a system of three linear equations in three dimensions, the result is three planes that may or may not intersect. The solution of a system of three equations in three variables can have one solution (the three planes intersect at a point), infinitely many solutions (the planes intersect at a line), or no solution (the planes do not intersect at the same point or line).

7. Solving by Elimination and/or Substitution
Take a few moments before beginning your calculations to examine the equations and make a plan for solving the system.
Use elimination or substitution to make a system of two equations with the same two variables from equations 1 & 2, 1 & 3, or 2 & 3.
Solve the systems to get the values of two of the variables.
Substitute your values in one of the original equations with three variables.

8. Solve the system of equations.
Step 1 Use elimination to make a system of two equations in two variables.
First equation
Second equation
Multiply by 2.
Add to eliminate z.
Example 5-1a

9. Subtract to eliminate z.
First equation
Third equation
Subtract to eliminate z.
Notice that the z terms in each equation have been eliminated. The result is two equations with the two same variables x and y.
Example 5-1a

10. Step 2 Solve the system of two equations.
Multiply by 5.
Add to eliminate y.
Divide by 29.
Substitute –2 for x in one of the two equations with two variables and solve for y.
Equation with two variables
Replace x with –2.
Multiply.
Simplify.
Example 5-1a

11. Equation with three variables
Step 3 Substitute –2 for x and 6 for y in one of the original equations with three variables.
Equation with three variables
Replace x with –2 and y with 6.
Multiply.
Simplify.
Answer: The solution is (–2, 6, –3). You can check this solution in the other two original equations.
Example 5-1a

12. Using a TI-84 Plus Calculator
Find the MATRIX button on your calculator. TI-84 Plus will press 2nd x-1 to get MATRIX
Scroll left once to highlight EDIT or twice to the right. Scroll down to choose a matrix other than A if needed, ENTER.
Make sure your Dimension is 3 x 4
Enter the coefficient and constant values
QUIT (2nd MODE)
Go back into MATRIX
Scroll right once to highlight MATH

13. Scroll down or up to highlight B: rref ( , ENTER.
Go back to MATRIX
Scroll down, if necessary to highlight your matrix that holds your data, ENTER.
ENTER again. The first three columns should contain only the numbers 1 and/or 0. Your ordered triple is the last column starting with your first variable.
If any row contains all zeros, then the system has infinite solutions. If a row does not have a 1 in the first three columns, but does have a value other than 0 in the last column then the system has no solution.
If your solutions are in decimal form, press MATH and enter twice to change the values to fraction form.

14. Solve the system of equations using a calculator.
Go to EDIT in the MATRIX menu and choose your matrix.
Enter the values 2, 3, −3, 16, 1, 1, 1, −3, 1, −2, −1, −1 while pressing ENTER after each value.
Exit and re-enter MATRIX. Choose MATH. Scroll to B: rref ( and press ENTER. Go back to MATRIX and choose the correct matrix, ENTER twice.
Answer: The last column contains the ordered triple (–1, 2, –4)
Example 5-1b

15. Solve the system of equations in a calculator.
Go to EDIT in the MATRIX menu and choose your matrix.
Enter the values 2, 1, -3, 5, 1, 2, -4, 7, 6, 3, -9, 15 while pressing ENTER after each value.
Exit and re-enter MATRIX. Choose MATH. Scroll down to B: rref ( and press ENTER. Go back to MATRIX and choose the correct matrix, ENTER twice.
Example 5-2a

16. Choose the MATH menu button and press ENTER twice
Choose the MATH menu button and press ENTER twice. This will change the values in the 3rd column from decimals to fractions. Notice that the 3rd row has all zeros. This indicates that the system has an infinite number of solutions.
Answer: The planes intersect in a line. So, there are an infinite number of solutions.

17. Solve the system of equations.
Answer: There are an infinite number of solutions.
Example 5-2b

18. Solve the system of equations.
Using the matrix function on the calculator, you should get the results of:
Answer: The last row has zeros in the first three columns and a value other than zero in the 4th column. This indicates there is no solution.

19. Solve the system of equations.
Answer: There is no solution of this system.
Example 5-3b

20. Explore Read the problem and define the variables.
Sports There are 49,000 seats in a sports stadium. Tickets for the seats in the upper level sell for $25, the ones in the middle level cost $30, and the ones in the bottom level are $35 each. The number of seats in the middle and bottom levels together equals the number of seats in the upper level. When all of the seats are sold for an event, the total revenue is $1,419,500. How many seats are there in each level?
Explore Read the problem and define the variables.
Example 5-4a

21. Plan There are 49,000 seats.
When all the seats are sold, the revenue is 1,419,500. Seats cost $25, $30, and $35.
The number of seats in the middle and bottom levels together equal the number of seats in the upper level.
– u + m + b = 0
Example 5-4a

22. Enter the values into a 3 x 4 matrix. The matrix should read:
Answer: There are 24,500 upper level, 10,100 middle level, and 14,400 bottom level seats.

23. Examine Check to see if all the criteria are met.
There are 49,000 seats in the stadium.
The number of seats in the middle and bottom levels equals the number of seats in the upper level.
When all of the seats are sold, the revenue is $1,419,500.
24,500($25) + 10,100($30) + 14,400($35) = $1,419,500
Example 5-4a

24. Answer: 10 pens, 7 pencils, 8 packs of paper
Business The school store sells pens, pencils, and paper. The pens are $1.25 each, the pencils are $0.50 each, and the paper is $2 per pack. Yesterday the store sold 25 items and earned $32. The number of pens sold equaled the number of pencils sold plus the number of packs of paper sold minus 5. How many of each item did the store sell?
Answer: 10 pens, 7 pencils, 8 packs of paper
Example 5-4b

25. Essential Question
How do you solve a system of equations in three variables using a calculator?
Use the MATRIX menu to enter the coefficients and constant values of each equation. Choose B: rref ( and the related matrix to determine the values of the variables. The values are located in the last column of the final matrix.

26. Teacher: Why is your homework paper blank?
Student: I solved all the systems by eliminating them.