Question 174551
1. Write a system of equations to represent the given information and solve using the substitution method. 
Please help and try to show every bit of work that you can to help me better understand the problem, Thank You! =)
:
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
:
{{{(matrix(3,3,1,1,1,1,-1,0,0,-4,1))}}}...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{{{(matrix(2,2,-1,0,-4,1))}}}-(1)det{{{(matrix(2,2,1,1,-4,1))}}}+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{{{(matrix(2,2,-1,0,-4,1))}}}-(-2)det{{{(matrix(2,2,1,1,-4,1))}}}
=40(-1)+2(5)=-30
:
s=det s/-6={{{highlight(-30/-6=5)}}}hours 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{{{(matrix(2,2,1,40,1,-2))}}}=1(-42)=-42
:
t=det t/-6={{{highlight(-42/-6=7)}}}hours 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){{{(matrix(2,2,1,40,1,-2))}}}+0=4(-42)=-168
:
c=det c/-6={{{highlight(-168/-6=28)}}}hours toy collecting