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11.2 Systems of Linear Equations: Three Variables

Eliminating one unknown at a time and back-substituting through three equations, and what parallel or coincident planes mean.

Algebra and Trigonometry 2e, 11.2 Systems of Linear Equations: Three Variables

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43 exercises

11.2 Exercise 1

Determine whether the ordered triples are solutions to the given system.

3x+y+z=2x+2y+z=-33x+y+2z=4\left\{\begin{matrix} 3 x + y + z = 2 \\ x + 2 y + z = - 3 \\ 3 x + y + 2 z = 4 \end{matrix}\right at (1, -3, 2)\left(1 , - 3 , 2\right) and at (4, -1, -5)\left(4 , - 1 , - 5\right)

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Determining Whether an Ordered Triple Is a Solution to a System

Determine whether the ordered triple ( 3,−2,1 ) \left(3 , −2 , 1\right) is a solution to the system.

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Solving a Real-World Problem Using a System of Three Equations in Three Variables

In the problem posed at the beginning of the section, Jordi invested his inheritance of $12,000 in three different funds: part in a money-market fund paying 3% interest annually; part in municipal bonds paying 4% annually; and the rest in mutual funds paying 7% annually. Jordi invested $4,000 more in mutual funds than he invested in municipal bonds. The total interest earned in one year was $670.…

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11.2 Exercise 10

Solve the system of equations in three variables.

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Solving an Inconsistent System of Three Equations in Three Variables

Solve the following system.

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11.2 Exercise 12

Solve the system of three equations in three variables.

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Finding the Solution to a Dependent System of Equations

Find the solution to the given system of three equations in three variables.

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11.2 Exercise 14

Solve the following system.

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11.2 Exercise 15

Can a linear system of three equations have exactly two solutions? Explain why or why not

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11.2 Exercise 17

If a given ordered triple does not solve the system of equations, is there no solution? If so, explain why. If not, give an example.

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11.2 Exercise 19

Can you explain whether there can be only one method to solve a linear system of equations? If yes, give an example of such a system of equations. If not, explain why not.

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11.2 Exercise 21

Determine whether the ordered triple given is the solution to the system of equations.

6x−y+3z=6 3x+5y+2z=0 x+y=0 \begin{matrix} 6 x - y + 3 z = 6 \\ 3 x + 5 y + 2 z = 0 \\ x + y = 0\end{matrix} and (3,−3,−5) \left(3 , −3 , −5\right)

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11.2 Exercise 23

Determine whether the ordered triple given is the solution to the system of equations.

x−y=0 x−z=5 x−y+z=−1 \begin{matrix}x - y = 0 \\ x - z = 5 \\ x - y + z = −1 \end{matrix} and (4,4,−1) \left(4 , 4 , −1\right)

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11.2 Exercise 25

Solve each system by elimination.

3x−4y+2z=−15 2x+4y+z=16 2x+3y+5z=20 \begin{matrix}3 x −4 y + 2 z = −15 \\ 2 x + 4 y + z = 16 \\ 2 x + 3 y + 5 z = 20\end{matrix}

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11.2 Exercise 27

Solve each system by elimination.

5x+2y+4z=9 −3x+2y+z=10 4x−3y+5z=−3 \begin{matrix} 5 x + 2 y + 4 z = 9 \\ −3 x + 2 y + z = 10 \\ 4 x −3 y + 5 z = −3\end{matrix}

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11.2 Exercise 29

Solve each system by elimination.

5x−2y+3z=4 −4x+6y−7z=−1 3x+2y−z=4 \begin{matrix} 5 x −2 y + 3 z = 4 \\ −4 x + 6 y −7 z = −1 \\ 3 x + 2 y - z = 4\end{matrix}

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11.2 Exercise 31

Solve each system by Gaussian elimination.

2x−y+3z=17 −5x+4y−2z=−46 2y+5z=−7 \begin{matrix} 2 x - y + 3 z = 17 \\ −5 x + 4 y −2 z = −46 \\ 2 y + 5 z = −7\end{matrix}

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11.2 Exercise 33

Solve each system by Gaussian elimination.

2x+3y−6z=1 −4x−6y+12z=−2 x+2y+5z=10 \begin{matrix} 2 x + 3 y −6 z = 1 \\ −4 x −6 y + 12 z = −2 \\ x + 2 y + 5 z = 10\end{matrix}

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11.2 Exercise 35

Solve each system by Gaussian elimination.

2x+3y−4z=5 −3x+2y+z=11 −x+5y+3z=4 \begin{matrix} 2 x + 3 y −4 z = 5 \\ −3 x + 2 y + z = 11 \\ - x + 5 y + 3 z = 4\end{matrix}

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11.2 Exercise 37

Solve each system by Gaussian elimination.

x+y+z=14 2y+3z=−14 −16y−24z=−112 \begin{matrix} x + y + z = 14 \\ 2 y + 3 z = −14 \\ −16 y −24 z = −112\end{matrix}

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11.2 Exercise 39

Solve each system by Gaussian elimination.

x+y+z=0 2x−y+3z=0 x−z=0 \begin{matrix} x + y + z = 0 \\ 2 x - y + 3 z = 0 \\ x - z = 0\end{matrix}

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11.2 Exercise 41

Solve each system by Gaussian elimination.

x+y+z=0 2x−y+3z=0 x−z=1 \begin{matrix} x + y + z = 0 \\ 2 x - y + 3 z = 0 \\ x - z = 1\end{matrix}

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11.2 Exercise 43

Solve each system by Gaussian elimination.

6x−5y+6z=38 1 5 x− 1 2 y+ 3 5 z=1 −4x− 3 2 y−z=−74 \begin{matrix} 6 x −5 y + 6 z = 38 \\ \frac{1}{5} x - \frac{1}{2} y + \frac{3}{5} z = 1 \\ −4 x - \frac{3}{2} y - z = −74\end{matrix}

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11.2 Exercise 45

Solve each system by Gaussian elimination.

− 1 3 x− 1 2 y− 1 4 z= 3 4 − 1 2 x− 1 4 y− 1 2 z=2 − 1 4 x− 3 4 y− 1 2 z=− 1 2 \begin{matrix}- \frac{1}{3} x - \frac{1}{2} y - \frac{1}{4} z = \frac{3}{4} \\ - \frac{1}{2} x - \frac{1}{4} y - \frac{1}{2} z = 2 \\ - \frac{1}{4} x - \frac{3}{4} y - \frac{1}{2} z = - \frac{1}{2}\end{matrix}

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11.2 Exercise 47

Solve each system by Gaussian elimination.

4 5 x− 7 8 y+ 1 2 z=1 − 4 5 x− 3 4 y+ 1 3 z=−8 − 2 5 x− 7 8 y+ 1 2 z=−5 \begin{matrix} \frac{4}{5} x - \frac{7}{8} y + \frac{1}{2} z = 1 \\ - \frac{4}{5} x - \frac{3}{4} y + \frac{1}{3} z = −8 \\ - \frac{2}{5} x - \frac{7}{8} y + \frac{1}{2} z = −5\end{matrix}

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11.2 Exercise 49

Solve each system by Gaussian elimination.

− 1 4 x− 5 4 y+ 5 2 z=−5 − 1 2 x− 5 3 y+ 5 4 z= 55 12 − 1 3 x− 1 3 y+ 1 3 z= 5 3 \begin{matrix} \\ - \frac{1}{4} x - \frac{5}{4} y + \frac{5}{2} z = −5 \\ - \frac{1}{2} x - \frac{5}{3} y + \frac{5}{4} z = \frac{55}{12} \\ - \frac{1}{3} x - \frac{1}{3} y + \frac{1}{3} z = \frac{5}{3}\end{matrix}

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11.2 Exercise 51

Solve each system by Gaussian elimination.

0.1x−0.2y+0.3z=2 0.5x−0.1y+0.4z=8 0.7x−0.2y+0.3z=8 \begin{matrix}0.1 x −0.2 y + 0.3 z = 2 \\ 0.5 x −0.1 y + 0.4 z = 8 \\ 0.7 x −0.2 y + 0.3 z = 8\end{matrix}

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11.2 Exercise 53

Solve each system by Gaussian elimination.

1.1x+0.7y−3.1z=−1.79 2.1x+0.5y−1.6z=−0.13 0.5x+0.4y−0.5z=−0.07 \begin{matrix}1.1 x + 0.7 y −3.1 z = −1.79 \\ 2.1 x + 0.5 y −1.6 z = −0.13 \\ 0.5 x + 0.4 y −0.5 z = −0.07\end{matrix}

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11.2 Exercise 55

Solve each system by Gaussian elimination.

0.1x+0.2y+0.3z=0.37 0.1x−0.2y−0.3z=−0.27 0.5x−0.1y−0.3z=−0.03 \begin{matrix}0.1 x + 0.2 y + 0.3 z = 0.37 \\ 0.1 x −0.2 y −0.3 z = −0.27 \\ 0.5 x −0.1 y −0.3 z = −0.03\end{matrix}

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11.2 Exercise 57

Solve each system by Gaussian elimination.

0.5x+0.2y−0.3z=1 0.4x−0.6y+0.7z=0.8 0.3x−0.1y−0.9z=0.6 \begin{matrix}0.5 x + 0.2 y −0.3 z = 1 \\ 0.4 x −0.6 y + 0.7 z = 0.8 \\ 0.3 x −0.1 y −0.9 z = 0.6\end{matrix}

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11.2 Exercise 59

Solve each system by Gaussian elimination.

0.8x+0.8y+0.8z=2.4 0.3x−0.5y+0.2z=0 0.1x+0.2y+0.3z=0.6 \begin{matrix}0.8 x + 0.8 y + 0.8 z = 2.4 \\ 0.3 x −0.5 y + 0.2 z = 0 \\ 0.1 x + 0.2 y + 0.3 z = 0.6\end{matrix}

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11.2 Exercise 61

Solve the system for x,y, x , y , and z. z .

5x−3y− z+1 2 = 1 2 6x+ y−9 2 +2z=−3 x+8 2 −4y+z=4 \begin{matrix}5 x −3 y - \frac{z + 1}{2} = \frac{1}{2} \\ 6 x + \frac{y −9}{2} + 2 z = −3 \\ \frac{x + 8}{2} −4 y + z = 4\end{matrix}

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11.2 Exercise 63

Solve the system for x,y, x , y , and z. z .

x−3 6 + y+2 2 − z−3 3 =2 x+2 4 + y−5 2 + z+4 2 =1 x+6 2 − y−3 2 +z+1=9 \begin{matrix}\frac{x −3}{6} + \frac{y + 2}{2} - \frac{z −3}{3} = 2 \\ \frac{x + 2}{4} + \frac{y −5}{2} + \frac{z + 4}{2} = 1 \\ \frac{x + 6}{2} - \frac{y −3}{2} + z + 1 = 9\end{matrix}

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11.2 Exercise 65

Three even numbers sum up to 108. The smaller is half the larger and the middle number is 3 4 \frac{3}{4} the larger. What are the three numbers?

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11.2 Exercise 67

At a family reunion, there were only blood relatives, consisting of children, parents, and grandparents, in attendance. There were 400 people total. There were twice as many parents as grandparents, and 50 more children than parents. How many children, parents, and grandparents were in attendance?

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11.2 Exercise 69

Your roommate, Shani, offered to buy groceries for you and your other roommate. The total bill was $82. She forgot to save the individual receipts but remembered that your groceries were $0.05 cheaper than half of her groceries, and that your other roommate’s groceries were $2.10 more than your groceries. How much was each of your share of the groceries?

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11.2 Exercise 71

Three coworkers work for the same employer. Their jobs are warehouse manager, office manager, and truck driver. The sum of the annual salaries of the warehouse manager and office manager is $82,000. The office manager makes $4,000 more than the truck driver annually. The annual salaries of the warehouse manager and the truck driver total $78,000. What is the annual salary of each of the co-workers…

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11.2 Exercise 73

A local band sells out for their concert. They sell all 1,175 tickets for a total purse of $28,112.50. The tickets were priced at $20 for student tickets, $22.50 for children, and $29 for adult tickets. If the band sold twice as many adult as children tickets, how many of each type was sold?

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11.2 Exercise 75

Last year, at Haven’s Pond Car Dealership, for a particular model of BMW, Jeep, and Toyota, one could purchase all three cars for a total of $140,000. This year, due to inflation, the same cars would cost $151,830. The cost of the BMW increased by 8%, the Jeep by 5%, and the Toyota by 12%. If the price of last year’s Jeep was $7,000 less than the price of last year’s BMW, what was the price of eac…

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11.2 Exercise 77

You inherit one million dollars. You invest it all in three accounts for one year. The first account pays 3% compounded annually, the second account pays 4% compounded annually, and the third account pays 2% compounded annually. After one year, you earn $34,000 in interest. If you invest four times the money into the account that pays 3% compared to 2%, how much did you invest in each account?

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11.2 Exercise 79

The top three countries in oil consumption in a certain year are as follows: the United States, Japan, and China. In millions of barrels per day, the three top countries consumed 39.8% of the world’s consumed oil. The United States consumed 0.7% more than four times China’s consumption. The United States consumed 5% more than triple Japan’s consumption. What percent of the world oil consumption di…

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11.2 Exercise 81

The top three sources of oil imports for the United States in the same year were Saudi Arabia, Mexico, and Canada. The three top countries accounted for 47% of oil imports. The United States imported 1.8% more from Saudi Arabia than they did from Mexico, and 1.7% more from Saudi Arabia than they did from Canada. What percent of the United States oil imports were from these three countries?“Oil res…

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11.2 Exercise 83

At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and fish, which comprised 55% of the endangered species. Birds accounted for 0.7% more than fish, and fish accounted for 1.5% more than mammals. What percent of the endangered species came from mammals, birds, and fish?

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.