1. Solving Linear Systems in Three Variables 3-6
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 2
Holt Algebra 2

2. Warm Up Solve each system of equations algebraically.
Classify each system and determine the number of solutions.
x = 4y + 10
4x + 2y = 4
6x – 5y = 9
2x – y =1 1.
(2, –2) 2.
(–1,–3)
3x – y = 8
6x – 2y = 2
x = 3y – 1
6x – 12y = –4 3. 4.
inconsistent; none
consistent, independent; one

3. Objectives
Represent solutions to systems of equations in three dimensions graphically.
Solve systems of equations in three dimensions algebraically.

4. Systems of three equations with three variables are often called 3-by-3 systems. In general, to find a single solution to any system of equations, you need as many equations as you have variables.

5. Recall from Lesson 3-5 that the graph of a linear equation in three variables 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 to the system is the set of points where all three planes intersect. These systems may have one, infinitely many, or no solution.

8. Identifying the exact solution from a graph of a 3-by-3 system can be very difficult. However, you can use the methods of elimination and substitution to reduce a 3-by-3 system to a 2-by-2 system and then use the methods that you learned in Lesson 3-2.

9. Example 1: Solving a Linear System in Three Variables
Use elimination to solve the system of equations.
5x – 2y – 3z = –7 1
2x – 3y + z = –16 2
3x + 4y – 2z = 7 3
Step 1 Eliminate one variable.
In this system, z is a reasonable choice to eliminate first because the coefficient of z in the second equation is 1 and z is easy to eliminate from the other equations.

10. Use equations and to create a second equation in x and y.
Example 1 Continued
5x – 2y – 3z = –7
5x – 2y – 3z = –7 1
Multiply equation by 3, and add to equation . 1 2
3(2x –3y + z = –16) 2
6x – 9y + 3z = –48
11x – 11y = –55 4
Use equations and to create a second equation in x and y. 3 2 1
3x + 4y – 2z = 7
3x + 4y – 2z = 7 3
Multiply equation by 2, and add to equation . 3 2
2(2x –3y + z = –16)
4x – 6y + 2z = –32 2
7x – 2y = –25 5

11. You now have a 2-by-2 system. 7x – 2y = –25
Example 1 Continued
11x – 11y = –55 4
You now have a 2-by-2 system.
7x – 2y = –25 5

12. You can eliminate y by using methods from Lesson 3-2.
Example 1 Continued
Step 2 Eliminate another variable. Then solve for the remaining variable.
You can eliminate y by using methods from Lesson 3-2.
Multiply equation by –2, and equation - by 11 and add. 4 5
–2(11x – 11y = –55)
–22x + 22y = 110 4 1
11(7x – 2y = –25)
77x – 22y = –275 5
55x = –165 1
x = –3
Solve for x.

13. Step 3 Use one of the equations in your 2-by-2 system to solve for y.
Example 1 Continued
Step 3 Use one of the equations in your 2-by-2 system to solve for y.
11x – 11y = –55 4 1
11(–3) – 11y = –55
Substitute –3 for x. 1
y = 2
Solve for y.

14. Substitute –3 for x and 2 for y. 2(–3) – 3(2) + z = –16
Example 1 Continued
Step 4 Substitute for x and y in one of the original equations to solve for z.
2x – 3y + z = –16 2
Substitute –3 for x and 2 for y.
2(–3) – 3(2) + z = –16 1
z = –4
Solve for y. 1
The solution is (–3, 2, –4).

15. Use elimination to solve the system of equations.
Check It Out! Example 1
Use elimination to solve the system of equations.
–x + y + 2z = 7 1
2x + 3y + z = 1 2
–3x – 4y + z = 4 3
Step 1 Eliminate one variable.
In this system, z is a reasonable choice to eliminate first because the coefficient of z in the second equation is 1.

16. Check It Out! Example 1 Continued
–x + y + 2z = 7
–x + y + 2z = 7 1
Multiply equation by –2, and add to equation . 1 2
–2(2x + 3y + z = 1)
–4x – 6y – 2z = –2 2
–5x – 5y = 5 4
Use equations and to create a second equation in x and y. 1 3 1
–x + y + 2z = 7
–x + y + 2z = 7
Multiply equation by –2, and add to equation . 1 3 1
–2(–3x – 4y + z = 4)
6x + 8y – 2z = –8 3
5x + 9y = –1 5

17. Check It Out! Example 1 Continued
–5x – 5y = 5 4
You now have a 2-by-2 system.
5x + 9y = –1 5

18. Check It Out! Example 1 Continued
Step 2 Eliminate another variable. Then solve for the remaining variable.
You can eliminate x by using methods from Lesson 3-2.
–5x – 5y = 5 4
Add equation to equation 4 5
5x + 9y = –1 5
4y = 4
Solve for y. 1
y = 1

19. Step 3 Use one of the equations in your 2-by-2 system to solve for x.
Check It Out! Example 1
Step 3 Use one of the equations in your 2-by-2 system to solve for x.
–5x – 5y = 5 4
Substitute 1 for y.
–5x – 5(1) = 5 1
Solve for x.
–5x – 5 = 5
–5x = 10 1
x = –2

20. Substitute –2 for x and 1 for y. 2x +3y + z = 1
Check It Out! Example 1
Step 4 Substitute for x and y in one of the original equations to solve for z.
Substitute –2 for x and 1 for y.
2x +3y + z = 1 2
2(–2) +3(1) + z = 1
– z = 1
Solve for z. 1
z = 2 1
The solution is (–2, 1, 2).

21. You can also use substitution to solve a 3-by-3 system
You can also use substitution to solve a 3-by-3 system. Again, the first step is to reduce the 3-by-3 system to a 2-by-2 system.

22. Example 2: Business Application
The table shows the number of each type of ticket sold and the total sales amount for each night of the school play. Find the price of each type of ticket.
Orchestra
Mezzanine
Balcony
Total Sales
Fri
200 30 40
$1470
Sat
250 60 50
$1950
Sun
150
$1050

23. Write a system of equations to represent the data in the table.
Example 2 Continued
Step 1 Let x represent the price of an orchestra seat, y represent the price of a mezzanine seat, and z represent the present of a balcony seat.
Write a system of equations to represent the data in the table.
200x + 30y + 40z = 1470
Friday’s sales. 1
250x + 60y + 50z = 1950
Saturday’s sales. 2
150x + 30y = 1050
Sunday’s sales. 3
A variable is “missing” in the last equation; however, the same solution methods apply. Elimination is a good choice because eliminating z is straightforward.

24. Multiply equation by 5 and equation by –4 and add.
Example 2 Continued
Step 2 Eliminate z.
Multiply equation by 5 and equation by –4 and add. 1 2
5(200x + 30y + 40z = 1470)
1000x + 150y + 200z = 7350 1
–4(250x + 60y + 50z = 1950)
–1000x – 240y – 200z = –7800 2
y = 5
By eliminating z, due to the coefficients of x, you also eliminated x providing a solution for y.

25. Step 3 Use equation to solve for x.
Example 2 Continued
Step 3 Use equation to solve for x. 3
150x + 30y = 1050
Substitute 5 for y. 3
150x + 30(5) = 1050
Solve for x.
x = 6

26. Step 4 Use equations or to solve for z.
Example 2 Continued
Step 4 Use equations or to solve for z. 2 1 1
200x + 30y + 40z = 1470
Substitute 6 for x and 5 for y. 1
200(6) + 30(5) + 40z = 1470
Solve for x.
z = 3
The solution to the system is (6, 5, 3). So, the cost of an orchestra seat is $6, the cost of a mezzanine seat is $5, and the cost of a balcony seat is $3.

27. Winter Fair Chili Cook-off
Check It Out! Example 2
Jada’s chili won first place at the winter fair. The table shows the results of the voting.
How many points are first-, second-, and third-place votes worth?
Winter Fair Chili Cook-off
Name
1st
Place
2nd
3rd
Total
Points
Jada 3 1 4 15
Maria 2 14 Al 13

28. Check It Out! Example 2 Continued
Step 1 Let x represent first-place points, y represent second-place points, and z represent third place points.
Write a system of equations to represent the data in the table.
3x + y + 4z = 15
Jada’s points. 1
2x + 4y = 14
Maria’s points. 2
2x + 2y + 3z = 13
Al’s points. 3
A variable is “missing” in one equation; however, the same solution methods apply. Elimination is a good choice because eliminating z is straightforward.

29. Check It Out! Example 2 Continued
Step 2 Eliminate z.
Multiply equation by 3 and equation by –4 and add. 3 1
3(3x + y + 4z = 15)
9x + 3y + 12z = 45 1
–4(2x + 2y + 3z = 13)
–8x – 8y – 12z = –52 3
x – 5y = –7 4
Multiply equation by –2 and add to equation 2 4
–2(x – 5y = –7)
–2x + 10y = 14 4
2x + 4y = 14
2x + 4y = 14 2
y = 2
Solve for y.

30. Check It Out! Example 2 Continued
Step 3 Use equation to solve for x. 2
2x + 4y = 14
Substitute 2 for y. 2
2x + 4(2) = 14
Solve for x.
x = 3

31. Check It Out! Example 2 Continued
Step 4 Substitute for x and y in one of the original equations to solve for z.
2x + 2y + 3z = 13 3
2(3) + 2(2) + 3z = 13
z = 13
z = 1
Solve for z.
The solution to the system is (3, 2, 1). The points for first-place is 3, the points for second-place is 2, and 1 point for third-place.

32. The systems in Examples 1 and 2 have unique solutions
The systems in Examples 1 and 2 have unique solutions. However, 3-by-3 systems may have no solution or an infinite number of solutions.
Consistent means that the system of equations has at least one solution.
Remember!

33. Example 3: Classifying Systems with Infinite Many Solutions or No Solutions
Classify the system as consistent or inconsistent, and determine the number of solutions.
2x – 6y + 4z = 2 1
–3x + 9y – 6z = –3 2
5x – 15y + 10z = 5 3

34. Example 3 Continued
The elimination method is convenient because the numbers you need to multiply the equations are small.
First, eliminate x.
Multiply equation by 3 and equation by 2 and add. 2 1
3(2x – 6y + 4z = 2)
6x – 18y + 12z = 6 1
2(–3x + 9y – 6z = –3)
–6x + 18y – 12z = –6 2
0 = 0 

35. Example 3 Continued
Multiply equation by 5 and equation by –2 and add. 3 1
5(2x – 6y + 4z = 2)
10x – 30y + 20z = 10 1
–2(5x – 15y + 10z = 5)
–10x + 30y – 20z = –10 3 
0 = 0
Because 0 is always equal to 0, the equation is an identity. Therefore, the system is consistent, dependent and has an infinite number of solutions.

36. Classify the system, and determine the number of solutions.
Check It Out! Example 3a
Classify the system, and determine the number of solutions.
3x – y + 2z = 4 1
2x – y + 3z = 7 2
–9x + 3y – 6z = –12 3

37. Check It Out! Example 3a Continued
The elimination method is convenient because the numbers you need to multiply the equations by are small.
First, eliminate y.
Multiply equation by –1 and add to equation 1 2
3x – y + 2z = 4
3x – y + 2z = 4 1
–1(2x – y + 3z = 7)
–2x + y – 3z = –7 3
x – z = –3 4

38. Check It Out! Example 3a Continued
Multiply equation by 3 and add to equation 3 2
3(2x – y + 3z = 7)
6x – 3y + 9z = 21 2
–9x + 3y – 6z = –12
–9x + 3y – 6z = –12 3
–3x z = 9 5
Now you have a 2-by-2 system.
x – z = –3 4
–3x + 3z = 9 5

39. Check It Out! Example 3a Continued
Eliminate x.
3(x – z = –3)
3x – 3z = –9 4
–3x + 3z = 9
–3x + 3z = 9 5
0 = 0 
Because 0 is always equal to 0, the equation is an identity. Therefore, the system is consistent, dependent, and has an infinite number of solutions.

40. Classify the system, and determine the number of solutions.
Check It Out! Example 3b
Classify the system, and determine the number of solutions.
2x – y + 3z = 6 1
2x – 4y + 6z = 10 2
y – z = –2 3

41. Check It Out! Example 3b Continued
Use the substitution method. Solve for y in equation 3.
y – z = –2
Solve for y. 3
y = z – 2 4
Substitute equation in for y in equation . 4 1
2x – y + 3z = 6
2x – (z – 2) + 3z = 6
2x – z z = 6
2x + 2z = 4 5

42. Check It Out! Example 3b Continued
Substitute equation in for y in equation . 4 2
2x – 4y + 6z = 10
2x – 4(z – 2) + 6z = 10
2x – 4z z = 10
2x + 2z = 2 6
Now you have a 2-by-2 system.
2x + 2z = 4 5
2x + 2z = 2 6

43. Check It Out! Example 3b Continued
Eliminate z.
2x + 2z = 4 5
–1(2x + 2z = 2) 6
0  2 
Because 0 is never equal to 2, the equation is a contradiction. Therefore, the system is inconsistent and has no solutions.

44. Hal 3 4 1 $17 Ina 2 5 $15 Joy $20 Lesson Quiz: Part I 1.
At the library book sale, each type of book is priced differently. The table shows the number of books Joy and her friends each bought, and the amount each person spent. Find the price of each type of book.
Hard-cover
Paper- back
Audio Books
Total Spent
Hal 3 4 1
$17
Ina 2 5
$15
Joy
$20
hardcover: $3;
paperback: $1;
audio books: $4

45. Lesson Quiz: Part II
Classify each system and determine the number of solutions.
2x – y + 2z = 5 2.
–3x +y – z = –1
inconsistent; none
x – y + 3z = 2
9x – 3y + 6z = 3 3.
12x – 4y + 8z = 4
consistent; dependent; infinite
–6x + 2y – 4z = 5