MATH 166 Spring 2016
4.3
c
Wen
Liu
4.3 Gauss Elimination for Systems of Linear Equations
Example of Gauss Elimination: Solve the following system of equations:
x + 2y = 20
y + 3x = 30
Elementary Equation Operations:
• Two equations can be interchanged.
• An equation may be multiplied by a non-zero constant.
• A multiple of one equation may be added to another equation.
Technology Corner: To solve a system using your calculator:
• To access MATRIX, press 2ND—x−1 —right arrow to the EDIT command— 1
• Enter the size, say 3 × 4, of the matrix you want to compute: 3 — ENTER — 4 and enter the
matrix elements and press ENTER . Press 2ND— MODE
• Press 2ND—x−1 —right arrow to the MATH command—scroll to B:rref(—- ENTER —2ND—
x−1 — 1 — ) — ENTER .
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MATH 166 Spring 2016
4.3
c
Wen
Liu
Examples:
1. A person has four times as many pennies as quarters. If the total face value of these coins is
$3.77, how many of each type of coin does this person have?
2. A t-shirt manufacturer makes three types of t-shirts: sleeveless, short-sleeve, and long-sleeve.
The time (in minutes) required by each department to produce a dozen t-shirts of each type is
shown in the following table. The cutting, sewing, and packaging departments have available a
maximum of 66, 136, and 40 labor-hours, respectively, per day. How many dozens of each type
of t-shirt can be produced each day if the plant is operated at full capacity?
Cutting
Sewing
Packaging
Sleeveless Short-Sleeve Long-Sleeve
9
12
15
22
24
28
6
8
8
3. For the opening night at the Opera House, a total of 1000 tickets were sold. Front orchestra
seats cost $100 apiece, rear orchestra seats cost $80 apiece, and front balcony seats cost $60
apiece. The combined number of tickets sold for the front orchestra and rear orchestra was 400
more than twice the number of front balcony tickets sold. The total receipts for the performance
were $83200. Determine how many tickets of each type were sold.
4. Cantwell Associates, a real estate developer, is planning to build a new apartment complex
consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 168 units
is planned, and the number of family units (two- and three-bedroom townhouses) will equal the
number of one-bedroom units. If the number of one-bedroom units will be 3 times the number
of three-bedroom units, find how many units of each type will be in the complex.
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MATH 166 Spring 2016
4.3
c
Wen
Liu
5. A private investment club has $200000 earmarked for investment in stocks. To arrive at an
acceptable overall level of risk, the stocks that management is considering have been classified
into three categories: high-risk, medium-risk, and low-risk. Management estimates that highrisk stocks will have a rate of return of 14%/year; medium-risk stocks, 9%/year; and low-risk
stocks, 5%/year. The members have decided that the investment in low-risk stocks should be
equal to the sum of the investments in the stocks of the other two categories. Determine how
much the club should invest in each type of stock if the investment goal is to have a return of
$18000/year on the total investment. (Assume that all the money available for investment is
invested.)
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