Stewart Precalc 6e Section 10.2: Systems of Linear Equations in Several Variables

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Stewart Precalc 6e Section 10.2: Systems of Linear Equations in Several Variables 0/52
1 Concepts · Level 1
Consider the system \(\begin{cases} x - y + z = 2 \\ -x + 2y + z = -3 \\ 3x + y - 2z = 2 \end{cases}\) If we add 2 times the first equation to the second equation, the second equation becomes _____ = _____.
2 Concepts · Level 1
Consider the system \(\begin{cases} x - y + z = 2 \\ -x + 2y + z = -3 \\ 3x + y - 2z = 2 \end{cases}\) To eliminate \(x\) from the third equation, we add _____ times the first equation to the third equation. The third equation becomes _____.
3 Skills - Identifying Linear Systems · Level 1
State whether the equation is linear. \(6x - \sqrt{3} y + \dfrac{1}{2} z = 0\)
4 Skills - Identifying Linear Systems · Level 1
State whether the equation is linear. \(x^2 + y^2 + z^2 = 4\)
5 Skills - Identifying Linear Systems · Level 1
State whether the system of equations is linear. \(\begin{cases} x y - 3y + z = 5 \\ x - y^2 + 5z = 0 \\ 2x + y z = 3 \end{cases}\)
6 Skills - Identifying Linear Systems · Level 1
State whether the system of equations is linear. \(\begin{cases} x - 2y + 3z = 10 \\ 2x + 5y = 2 \\ y + 2z = 4 \end{cases}\)
7 Skills - Triangular System · Level 2
Use back-substitution to solve the triangular system. \(\begin{cases} x - 2y + 4z = 3 \\ y + 2z = 7 \\ z = 2 \end{cases}\)
8 Skills - Triangular System · Level 2
Use back-substitution to solve the triangular system. \(\begin{cases} x + y - 3z = 8 \\ y - 3z = 5 \\ z = -1 \end{cases}\)
9 Skills - Triangular System · Level 2
Use back-substitution to solve the triangular system. \(\begin{cases} x + 2y + z = 7 \\ -y + 3z = 9 \\ 2z = 6 \end{cases}\)
10 Skills - Triangular System · Level 2
Use back-substitution to solve the triangular system. \(\begin{cases} x - 2y + 3z = 10 \\ 2y - z = 2 \\ 3z = 12 \end{cases}\)
11 Skills - Triangular System · Level 2
Use back-substitution to solve the triangular system. \(\begin{cases} 2x - y + 6z = 5 \\ y + 4z = 0 \\ -2z = 1 \end{cases}\)
12 Skills - Triangular System · Level 2
Use back-substitution to solve the triangular system. \(\begin{cases} 4x + 3z = 10 \\ 2y - z = -6 \\ \dfrac{1}{2} z = 4 \end{cases}\)
13 Skills - Elimination Operations · Level 2
Perform an operation on the system that eliminates \(x\) from the second equation. Write the new equivalent system. \(\begin{cases} x - 2y - z = 4 \\ x - y + 3z = 0 \\ 2x + y + z = 0 \end{cases}\)
14 Skills - Elimination Operations · Level 2
Perform an operation on the system that eliminates \(x\) from the second equation. Write the new equivalent system. \(\begin{cases} x + y - 3z = 3 \\ -2x + 3y + z = 2 \\ x - y + 2z = 0 \end{cases}\)
15 Skills - Elimination Operations · Level 2
Perform an operation on the system that eliminates \(x\) from the third equation. Write the new equivalent system. \(\begin{cases} 2x - y + 3z = 2 \\ x + 2y - z = 4 \\ -4x + 5y + z = 10 \end{cases}\)
16 Skills - Elimination Operations · Level 2
Perform an operation on the system that eliminates \(y\) from the third equation. Write the new equivalent system. \(\begin{cases} x - 4y + z = 3 \\ y - 3z = 10 \\ 3y - 8z = 24 \end{cases}\)
17 Skills - Solve Linear System · Level 3
\( \begin{cases} x - y - z = 4 \\ 2y + z = -1 \\ -x + y - 2z = 5 \end{cases} \)
18 Skills - Solve Linear System · Level 3
\( \begin{cases} x - y + z = 0 \\ y + 2z = -2 \\ x + y - z = 2 \end{cases} \)
19 Skills - Solve Linear System · Level 3
\( \begin{cases} x + y + z = 4 \\ x + 3y + 3z = 10 \\ 2x + y - z = 3 \end{cases} \)
20 Skills - Solve Linear System · Level 3
\( \begin{cases} x + y + z = 0 \\ -x + 2y + 5z = 3 \\ 3x - y = 6 \end{cases} \)
21 Skills - Solve Linear System · Level 3
\( \begin{cases} x - 4z = 1 \\ 2x - y - 6z = 4 \\ 2x + 3y - 2z = 8 \end{cases} \)
22 Skills - Solve Linear System · Level 3
\( \begin{cases} x - y + 2z = 2 \\ 3x + y + 5z = 8 \\ 2x - y - 2z = -7 \end{cases} \)
23 Skills - Solve Linear System · Level 3
\( \begin{cases} 2x + 4y - z = 2 \\ x + 2y - 3z = -4 \\ 3x - y + z = 1 \end{cases} \)
24 Skills - Solve Linear System · Level 3
\( \begin{cases} 2x + y - z = -8 \\ -x + y + z = 3 \\ -2x + 4z = 18 \end{cases} \)
25 Skills - Solve Linear System · Level 3
\( \begin{cases} y - 2z = 0 \\ 2x + 3y = 2 \\ -x - 2y + z = -1 \end{cases} \)
26 Skills - Solve Linear System · Level 3
\( \begin{cases} 2y + z = 3 \\ 5x + 4y + 3z = -1 \\ x - 3y = -2 \end{cases} \)
27 Skills - Solve Linear System · Level 3
\( \begin{cases} x + 2y - z = 1 \\ 2x + 3y - 4z = -3 \\ 3x + 6y - 3z = 4 \end{cases} \)
28 Skills - Solve Linear System · Level 3
\( \begin{cases} -x + 2y + 5z = 4 \\ x - 2z = 0 \\ 4x - 2y - 11z = 2 \end{cases} \)
29 Skills - Solve Linear System · Level 3
\( \begin{cases} 2x + 3y - z = 1 \\ x + 2y = 3 \\ x + 3y + z = 4 \end{cases} \)
30 Skills - Solve Linear System · Level 3
\( \begin{cases} x - 2y - 3z = 5 \\ 2x + y - z = 5 \\ 4x - 3y - 7z = 5 \end{cases} \)
31 Skills - Solve Linear System · Level 3
\( \begin{cases} x + y - z = 0 \\ x + 2y - 3z = -3 \\ 2x + 3y - 4z = -3 \end{cases} \)
32 Skills - Solve Linear System · Level 3
\( \begin{cases} x - 2y + z = 3 \\ 2x - 5y + 6z = 7 \\ 2x - 3y - 2z = 5 \end{cases} \)
33 Skills - Solve Linear System · Level 3
\( \begin{cases} x + 3y - 2z = 0 \\ 2x + 4z = 4 \\ 4x + 6y = 4 \end{cases} \)
34 Skills - Solve Linear System · Level 3
\( \begin{cases} 2x + 4y - z = 3 \\ x + 2y + 4z = 6 \\ x + 2y - 2z = 0 \end{cases} \)
35 Skills - Solve Linear System · Level 4
\( \begin{cases} x + z + 2w = 6 \\ y - 2z = -3 \\ x + 2y - z = -2 \\ 2x + y + 3z - 2w = 0 \end{cases} \)
36 Skills - Solve Linear System · Level 4
\( \begin{cases} x + y + z + w = 0 \\ x + y + 2z + 2w = 0 \\ 2x + 2y + 3z + 4w = 1 \\ 2x + 3y + 4z + 5w = 2 \end{cases} \)
37 Applications - Finance · Level 4
An investor has \$100,000 to invest in three types of bonds: short-term, intermediate-term, and long-term. Short-term bonds pay 4% annually, intermediate-term bonds pay 5%, and long-term bonds pay 6%. The investor wishes to realize a total annual income of 5.1%, with equal amounts invested in short- and intermediate-term bonds. How much should she invest in each type?
38 Applications - Finance · Level 4
An investor has \$100,000 to invest in three types of bonds: short-term, intermediate-term, and long-term. Short-term bonds pay 4% annually, intermediate-term bonds pay 6%, and long-term bonds pay 8%. The investor wishes to have a total annual return of \$6700 on her investment, with equal amounts invested in intermediate- and long-term bonds. How much should she invest in each type?
39 Applications - Agriculture · Level 4
A farmer has 1200 acres of land on which he grows corn, wheat, and soybeans. It costs \$45 per acre to grow corn, \$60 to grow wheat, and \$50 to grow soybeans. Because of market demand, the farmer will grow twice as many acres of wheat as of corn. He has allocated \$63,750 for the cost of growing his crops. How many acres of each crop should he plant?
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40 Applications - Gas Station · Level 4
A gas station sells three types of gas: Regular for \$3.00 a gallon, Performance Plus for \$3.20 a gallon, and Premium for \$3.30 a gallon. On a particular day 6500 gallons of gas were sold for a total of \$20,050. Three times as many gallons of Regular as Premium gas were sold. How many gallons of each type of gas were sold that day?
41 Applications - Nutrition · Level 4
A biologist wants to feed each rabbit a diet that contains exactly 9 mg of niacin, 14 mg of thiamin, and 32 mg of riboflavin. Three pellets are available with the following content per ounce:
Type A Type B Type C
Niacin (mg) 2 3 1
Thiamin (mg) 3 1 3
Riboflavin (mg) 8 5 7
How many ounces of each type of food should each rabbit be given daily?
42 Applications - Diet Program · Level 4
Nicole's diet requires each meal to have 460 calories, 6 grams of fiber, and 11 grams of fat. Per serving:
Food Fiber Fat Calories
Toast 2 1 100
Cottage cheese 0 5 120
Fruit 2 0 60
How many servings of each food should Nicole eat to follow her diet?
43 Applications - Juice Blends · Level 4
The Juice Company offers three smoothies. Each contains (oz):
Smoothie Mango Pineapple Orange
Midnight Mango 8 3 3
Tropical Torrent 6 5 3
Pineapple Power 2 8 4
On a particular day the company used 820 oz of mango juice, 690 oz of pineapple juice, and 450 oz of orange juice. How many smoothies of each kind were sold that day?
44 Applications - Appliance Manufacturing · Level 4
Kitchen Korner produces appliances at three factories, with the following daily production:
Appliance Factory A Factory B Factory C
Refrigerators 8 10 14
Dishwashers 16 12 10
Stoves 10 18 6
Kitchen Korner receives an order for 110 refrigerators, 150 dishwashers, and 114 stoves. How many days should each plant be scheduled to fill this order?
45 Applications - Stock Portfolio · Level 4
An investor owns three stocks: A, B, and C. The closing prices on three successive trading days are:
Stock A Stock B Stock C
Monday \$10 \$25 \$29
Tuesday \$12 \$20 \$32
Wednesday \$16 \$15 \$32
Despite the volatility, the total value of the stocks remained \$74,000 at the end of each day. How many shares of each stock does the investor own?
46 Applications - Electricity · Level 4
By using Kirchhoff's Laws, the currents \(I_1\), \(I_2\), and \(I_3\) that pass through the three branches of the circuit shown in the figure satisfy the following linear system. Solve the system to find \(I_1\), \(I_2\), and \(I_3\). \(\begin{cases} I_1 + I_2 - I_3 = 0 \\ 16 I_1 - 8 I_2 = 4 \\ 8 I_2 + 4 I_3 = 5 \end{cases}\)
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47 Discovery, Discussion, Writing · Level 5
Can a linear system have exactly two solutions?
(a) Suppose that \((x_0, y_0, z_0)\) and \((x_1, y_1, z_1)\) are solutions of the system \(\begin{cases} a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3 \end{cases}\) Show that \(((x_0 + x_1)/2, (y_0 + y_1)/2, (z_0 + z_1)/2)\) is also a solution.
(b) Use the result of part (a) to prove that if the system has two different solutions, then it has infinitely many solutions.

Enter your answer directly below each part above.

48 Example - Back-Substitution · Level 2
Solve the system using back-substitution. \(\begin{cases} x - 2 y - z = 1 \\ y + 2 z = 5 \\ z = 3 \end{cases}\)
49 Example - Gaussian Elimination · Level 3
Solve the system using Gaussian elimination. \(\begin{cases} x - 2 y + 3 z = 1 \\ x + 2 y - z = 13 \\ 3 x + 2 y - 5 z = 3 \end{cases}\)
50 Example - System with No Solution · Level 3
Solve the system. \(\begin{cases} x + 2 y - 2 z = 1 \\ 2 x + 2 y - z = 6 \\ 3 x + 4 y - 3 z = 5 \end{cases}\)
51 Example - A System with Infinitely Many Solutions · Level 3
Solve the following system: \(\begin{cases} x - y + 5z = -2 \\ 2x + y + 4z = 2 \\ 2x + 4y - 2z = 8 \end{cases}\)
52 Example - Modeling a Financial Problem Using a Linear System · Level 3
Jason receives an inheritance of \$50,000. His financial advisor suggests that he invest this in three mutual funds: a money-market fund, a blue-chip stock fund, and a high-tech stock fund. The advisor estimates that the money-market fund will return 5% over the next year, the blue-chip fund 9%, and the high-tech fund 16%. Jason wants a total first-year return of \$4000. To avoid excessive risk, he decides to invest three times as much in the money-market fund as in the high-tech stock fund. How much should he invest in each fund?

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