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 โxinto the second equation to get .045x+
.08(3000 โx) = 177, which simplifies to โ.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=โ5 and xโ5y=โ8.
Answer. There are several possible methods to solve this. Using substitution, the second
equation implies x= 5yโ8, and we substitute 5yโ8 for xin the first equation to get
3(5yโ8) + 4y=โ5, or 19yโ24 = โ5, and so y= 1 and x= 5 โ8 = โ3.
Check: 3(โ3) + 4(1) = โ9 + 4 = โ5 and โ3โ5(1) = โ3โ5 = โ8, thus both equations are
satisfied by x=โ3 and y= 1.
3. Consider the triangular system of equations
xโ2y+z= 5
yโ3z=โ5
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 yโ6 = โ5, or y= 1, and then xโ2 + 2 = 5, or x= 5. Thus
the solution is x= 5, y= 1 and z= 2.