4.4E: Exercises


Practice Makes Perfect

Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables

In the following exercises, determine whether the ordered triple is a solution to the system.

1. $$\left\{ \begin{array} {l} 2x−6y+z=3 \\ 3x−4y−3z=2 \\ 2x+3y−2z=3 \end{array} \right.$$

ⓐ $$(3,1,3)$$
ⓑ $$(4,3,7)$$

2. $$\left\{ \begin{array} {l} -3x+y+z=-4 \\ -x+2y-2z=1 \\ 2x-y-z=-1 \end{array} \right.$$

ⓐ $$(−5,−7,4)$$
ⓑ $$(5,7,4)$$

ⓐ no ⓑ yes

3. $$\left\{ \begin{array} {l} y−10z=−8 \\ 2x−y=2 \\ x−5z=3 \end{array} \right.$$

ⓐ $$(7,12,2)$$
ⓑ $$(2,2,1)$$

4. $$\left\{ \begin{array} {l} x+3y−z=1 \\ 5y=\frac{2}{3}x \\ −2x−3y+z=−2 \end{array} \right.$$

ⓐ $$(−6,5,12)$$
ⓑ $$(5,\frac{4}{3},−3)$$

ⓐ no ⓑ yes

Solve a System of Linear Equations with Three Variables

In the following exercises, solve the system of equations.

5. $$\left\{ \begin{array} {l} 5x+2y+z=5 \\ −3x−y+2z=6 \\ 2x+3y−3z=5 \end{array} \right.$$

6. $$\left\{ \begin{array} {l} 6x−5y+2z=3 \\ 2x+y−4z=5 \\ 3x−3y+z=−1 \end{array} \right.$$

$$(4,5,2)$$

7. $$\left\{ \begin{array} {l} 2x−5y+3z=8 \\ 3x−y+4z=7 \\ x+3y+2z=−3 \end{array} \right.$$

8. $$\left\{ \begin{array} {l} 5x−3y+2z=−5 \\ 2x−y−z=4 \\ 3x−2y+2z=−7 \end{array} \right.$$

$$(7,12,−2)$$

9. $$\left\{ \begin{array} {l} 3x−5y+4z=5 \\ 5x+2y+z=0 \\ 2x+3y−2z=3 \end{array} \right.$$

10. $$\left\{ \begin{array} {l} 4x−3y+z=7 \\ 2x−5y−4z=3 \\ 3x−2y−2z=−7 \end{array} \right.$$

$$(−3,−5,4)$$

11. $$\left\{ \begin{array} {l} 3x+8y+2z=−5 \\ 2x+5y−3z=0 \\ x+2y−2z=−1 \end{array} \right.$$

12. $$\left\{ \begin{array} {l} 11x+9y+2z=−9 \\ 7x+5y+3z=−7 \\ 4x+3y+z=−3 \end{array} \right.$$

$$(2,−3,−2)$$

13. $$\left\{ \begin{array} {l} \frac{1}{3}x−y−z=1 \\ x+\frac{5}{2}y+z=−2 \\ 2x+2y+\frac{1}{2}z=−4 \end{array} \right.$$

14. $$\left\{ \begin{array} {l} x+\frac{1}{2}y+\frac{1}{2}z=0 \\ \frac{1}{5}x−\frac{1}{5}y+z=0 \\ \frac{1}{3}x−\frac{1}{3}y+2z=−1 \end{array} \right.$$

$$(6,−9,−3)$$

15. $$\left\{ \begin{array} {l} x+\frac{1}{3}y−2z=−1 \\ \frac{1}{3}x+y+\frac{1}{2}z=0 \\ \frac{1}{2}x+\frac{1}{3}y−\frac{1}{2}z=−1 \end{array} \right.$$

16. $$\left\{ \begin{array} {l} \frac{1}{3}x−y+\frac{1}{2}z=4 \\ \frac{2}{3}x+\frac{5}{2}y−4z=0 \\ x−\frac{1}{2}y+\frac{3}{2}z=2 \end{array} \right.$$

$$(3,−4,−2)$$

17. $$\left\{ \begin{array} {l} x+2z=0 \\ 4y+3z=−2 \\ 2x−5y=3 \end{array} \right.$$

18. $$\left\{ \begin{array} {l} 2x+5y=4 \\ 3y−z=\frac{3}{4} \\ x+3z=−3 \end{array} \right.$$

$$(−3,2,3)$$

19. $$\left\{ \begin{array} {l} 2y+3z=−1 \\ 5x+3y=−6 \\ 7x+z=1 \end{array} \right.$$

20. $$\left\{ \begin{array} {l} 3x−z=−3 \\ 5y+2z=−6 \\ 4x+3y=−8 \end{array} \right.$$

$$(−2,0,−3)$$

21. $$\left\{ \begin{array} {l} 4x−3y+2z=0 \\ −2x+3y−7z=1 \\ 2x−2y+3z=6 \end{array} \right.$$

22. $$\left\{ \begin{array} {l} x−2y+2z=1 \\ −2x+y−z=2 \\ x−y+z=5 \end{array} \right.$$

no solution

23. $$\left\{ \begin{array} {l} 2x+3y+z=1 \\ 2x+y+z=9 \\ 3x+4y+2z=20 \end{array} \right.$$

24. $$\left\{ \begin{array} {l} x+4y+z=−8 \\ 4x−y+3z=9 \\ 2x+7y+z=0 \end{array} \right.$$

$$x=\frac{203}{16};\space y=–\frac{25}{16};\space z=–\frac{231}{16};$$

25. $$\left\{ \begin{array} {l} x+2y+z=4 \\ x+y−2z=3 \\ −2x−3y+z=−7 \end{array} \right.$$

26. $$\left\{ \begin{array} {l} x+y−2z=3 \\ −2x−3y+z=−7 \\ x+2y+z=4 \end{array} \right.$$

$$(x,y,z)$$ where $$x=5z+2;\space y=−3z+1;\space z$$ is any real number

27. $$\left\{ \begin{array} {l} x+y−3z=−1 \\ y−z=0 \\ −x+2y=1 \end{array} \right.$$

28. $$\left\{ \begin{array} {l} x−2y+3z=1 \\ x+y−3z=7 \\ 3x−4y+5z=7 \end{array} \right.$$

$$(x,y,z)$$ where $$x=5z−2;\space y=4z−3;\space z$$ is any real number

Solve Applications using Systems of Linear Equations with Three Variables

In the following exercises, solve the given problem.

29. The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is twice the measure of the first angle. The third angle is twelve more than the second. Find the measures of the three angles.

30. The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is fifteen more than the second. Find the measures of the three angles.

42, 50, 58

31. After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases 4 t-shirts, the video, and 1 stuffed animal, their total is $135. A couple buys 2 t-shirts, the video, and 3 stuffed animals for their nieces and spends$115. Another couple buys 2 t-shirts, the video, and 1 stuffed animal and their total is $85. What is the cost of each item? 32. The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of$65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of $140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 cans of popcorn for a total sales of$250. What is the cost of each item?

$20,$5, \$10

Writing Exercises

33. In your own words explain the steps to solve a system of linear equations with three variables by elimination.

34. How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?