SOLUTION: As a participant in your school's community service project, you volunteer a total of 40 hours over the course of the school year. Your volunteer hours include serving at a soup k

Algebra ->  Matrices-and-determiminant -> SOLUTION: As a participant in your school's community service project, you volunteer a total of 40 hours over the course of the school year. Your volunteer hours include serving at a soup k      Log On


   

Question 174668This question is from textbook Algebra 2, Texas Edition
: As a participant in your school's community service project, you volunteer a total of 40 hours over the course of the school year. Your volunteer hours include serving at a soup kitchen, picking up trash at several local parks, and collecting toys for needy children. You spend 4 times as many hours collecting toys as picking up trash, and 2 hours less serving at the soup kitchen than picking up trash. Slove this system of equations using Cramer's Rule. Suppose you cannot remember the total number of hours you volunteered. From the remaining information, can you still determine how many hours you spend doing each volunteer activity? Why or why not? This question is from textbook Algebra 2, Texas Edition

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!

lets call the time spent at soup kitchen,trash pickup, and toy collecting
s, t, and c, respectively
:
c=4t...........eq 1
s=t-2..........eq 2
s+t+c=40.......eq 3
:
rewrite equations
:
s+t+c=40......eq 3
s-t+0c=-2.....eq 2
0s-4t+c=0......eq 1
:
now writing out the main matrix
:
%28matrix%283%2C3%2C1%2C1%2C1%2C1%2C-1%2C0%2C0%2C-4%2C1%29%29...lets call this matrix X
:
we need to find the determinant of this matrix to begin with
:
we can choose any row or column to determine this ..I will work with column 1 since it has a zero in it
:
det X=(1) det%28matrix%282%2C2%2C-1%2C0%2C-4%2C1%29%29-(1)det%28matrix%282%2C2%2C1%2C1%2C-4%2C1%29%29+0=
.....1(-1)-1(5)=-6
:
s=det s/-6 where matrix s is matrix X with the 1st column replaced by constant coefficents 40,-2,0. I will use column one for our determinant
:
det s = 40 det%28matrix%282%2C2%2C-1%2C0%2C-4%2C1%29%29-(-2)det%28matrix%282%2C2%2C1%2C1%2C-4%2C1%29%29
=40(-1)+2(5)=-30
:
s=det s/-6=highlight%28-30%2F-6=5%29hours in the soup kitchen
:
:
t=det t/-6 where matrix t is matrix X with the 2nd column replaced by constant coefficents 40,-2,0. I will use row 3 for our determinant as it has 2 zeros
:
det t=0+0+(1)det%28matrix%282%2C2%2C1%2C40%2C1%2C-2%29%29=1(-42)=-42
:
t=det t/-6=highlight%28-42%2F-6=7%29hours picking up trash
:
:
c=det c/-6 where matrix c is matrix X with the 3rd column replaced by constant coefficents 40,-2,0. I will use row 3 for our determinant as it has 2 zeros
:
det c= 0-(-4)%28matrix%282%2C2%2C1%2C40%2C1%2C-2%29%29+0=4(-42)=-168
:
c=det c/-6=highlight%28-168%2F-6=28%29hours toy collecting

If you forgot the number of hours you volunteered you could not determine the hours you spend doing each activity....
:
you would only have 2 equations and 3 unknowns