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Solved Questions for College Algebra - Quiz V | MATH 121, Quizzes of Algebra

Material Type: Quiz; Class: COLLEGE ALGEBRA; Subject: MATHEMATICS; University: La Sierra University; Term: Fall 2004;

Typology: Quizzes

Pre 2010

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Math 121, Quiz 5, November 5, 2004
Name: Answers and Hints
1. An investment of $3000 is placed in stocks and bonds. The annual rate of return on the
stocks is 4.5%, the annual rate of return on the bonds is 8%. The annual return from the
stocks and bonds is $177. Find the amount invested in each stocks and bonds.
Answer. Let xbe the amount invested in stocks and let ybe the amount invested in bonds.
Then
x+y= 3000 and .045x+.08y= 177.
Using the substitution method, we put y= 3000 โˆ’xinto the second equation to get .045x+
.08(3000 โˆ’x) = 177, which simplifies to โˆ’.035x+ 240 = 177, and so .035x= 63, or x=
63/.035 = 1800. Therefore, y= 1200. So, $1800 was invested in stocks, and $1200 was
invested in bonds.
Check: 4.5% of 1800 + 8% of 1200 is 81 + 96 = 177.
2. Solve the system of equations 3x+ 4y=โˆ’5 and xโˆ’5y=โˆ’8.
Answer. There are several possible methods to solve this. Using substitution, the second
equation implies x= 5yโˆ’8, and we substitute 5yโˆ’8 for xin the first equation to get
3(5yโˆ’8) + 4y=โˆ’5, or 19yโˆ’24 = โˆ’5, and so y= 1 and x= 5 โˆ’8 = โˆ’3.
Check: 3(โˆ’3) + 4(1) = โˆ’9 + 4 = โˆ’5 and โˆ’3โˆ’5(1) = โˆ’3โˆ’5 = โˆ’8, thus both equations are
satisfied by x=โˆ’3 and y= 1.
3. Consider the triangular system of equations
xโˆ’2y+z= 5
yโˆ’3z=โˆ’5
cz = 2
(a) For which value(s) of c, if any, does the system have infinitely many solutions?
(b) For which value(s) of c, if any, does the system have a unique solution?
(c) For which value(s) of c, if any, does the system have no solution?
(d) Find the solution to the system if c= 1.
Answer. (a) No values of clead to infinitely many solutions. All values of care accounted for
in the cases for (b) and (c) below.
(b) c6= 0
(c) c= 0, because then 0 = 2 which is impossible.
(d) If c= 1, then z= 2, and so yโˆ’6 = โˆ’5, or y= 1, and then xโˆ’2 + 2 = 5, or x= 5. Thus
the solution is x= 5, y= 1 and z= 2.

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Math 121, Quiz 5, November 5, 2004

Name: Answers and Hints

  1. An investment of $3000 is placed in stocks and bonds. The annual rate of return on the stocks is 4.5%, the annual rate of return on the bonds is 8%. The annual return from the stocks and bonds is $177. Find the amount invested in each stocks and bonds.

Answer. Let x be the amount invested in stocks and let y be the amount invested in bonds. Then x + y = 3000 and. 045 x +. 08 y = 177. Using the substitution method, we put y = 3000 โˆ’ x into the second equation to get. 045 x + .08(3000 โˆ’ x) = 177, which simplifies to โˆ’. 035 x + 240 = 177, and so. 035 x = 63, or x = 63 /.035 = 1800. Therefore, y = 1200. So, $1800 was invested in stocks, and $1200 was invested in bonds. Check: 4.5% of 1800 + 8% of 1200 is 81 + 96 = 177.

  1. Solve the system of equations 3 x + 4y = โˆ’5 and x โˆ’ 5 y = โˆ’8.

Answer. There are several possible methods to solve this. Using substitution, the second equation implies x = 5y โˆ’ 8, and we substitute 5y โˆ’ 8 for x in the first equation to get 3(5y โˆ’ 8) + 4y = โˆ’5, or 19y โˆ’ 24 = โˆ’5, and so y = 1 and x = 5 โˆ’ 8 = โˆ’3. Check: 3(โˆ’3) + 4(1) = โˆ’9 + 4 = โˆ’5 and โˆ’ 3 โˆ’ 5(1) = โˆ’ 3 โˆ’ 5 = โˆ’8, thus both equations are satisfied by x = โˆ’3 and y = 1.

  1. Consider the triangular system of equations x โˆ’ 2 y + z = 5 y โˆ’ 3 z = โˆ’ 5 cz = 2

(a) For which value(s) of c, if any, does the system have infinitely many solutions? (b) For which value(s) of c, if any, does the system have a unique solution? (c) For which value(s) of c, if any, does the system have no solution? (d) Find the solution to the system if c = 1.

Answer. (a) No values of c lead to infinitely many solutions. All values of c are accounted for in the cases for (b) and (c) below. (b) c 6 = 0 (c) c = 0, because then 0 = 2 which is impossible. (d) If c = 1, then z = 2, and so y โˆ’ 6 = โˆ’5, or y = 1, and then x โˆ’ 2 + 2 = 5, or x = 5. Thus the solution is x = 5, y = 1 and z = 2.