Express the situation as a system of two equations in two variables. Be sure to
ID: 3008075 • Letter: E
Question
Express the situation as a system of two equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by row-reducing the corresponding augmented matrix. State your final answer in terms of the original question.
For the final days before the election, the campaign manager has a total of $28,000 to spend on TV and radio campaign advertisements. Each TV ad costs $3000 and is seen by 10,000 voters, while each radio ad costs $500 and is heard by 2000 voters. Ignoring repeated exposures to the same voter, how many TV and radio ads will contact 102,000 voters using the allocated funds?
Explanation / Answer
Here we let that total tv ads be x and total radio ads be y, so by first condition
total cost of x tv ads + total cost of y radio ads = 28000
or 3000 x + 500 y= 28000
or 500(6x+y)= 28000
or 6x+y= 28000/500 = 56 ........(i)
Again total voters for tv ads + total voters for radio ads = 102000
or 10000 x + 2000 y = 102000
or 2000 (5x + y) = 102000
or 5x+ y= 102000/2000 = 51 ...........(ii)
Thus equation (i) and (ii) form the required system of two equations.
Now we put the matrix formed by coefficients of x and y in row reducing form and get its inverse as
[ 6 1] = [ 1 0] where A = [6 1]
[5 1] [0 1] A [5 1]
Now on R1=R1/6 , R2 = R2-5R1
[ 1 1/6] = [ 1/6 0]
[0 1-5/6=1/6] [-5/6 1] A
Now again on R2=6R2 , R1 = R1-1/6R2
[ 1 1/6-1/6] = [ 1/6+5/6 -1]
[0 1] [-5 6] A
or
[ 1 0] = [ 1 -1]
[0 1] [-5 6] A
so here A^(-1) = [ 1 -1]
[-5 6]
so by rule , we have
[x ] =[ 1 -1] [ 56]
[y] = [-5 6] [51]
or
[x] = [1(56)-1(51)] = [5]
[y] [-5(56)+6(51)] [26]
so clearly x= 5 and y= 26
that means total 5 TV ads and 26 radio ads will contact 102,000 voters using the allocated fund.
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