a.) Formulate the situation as a system of two linear equations in two variables

Question

a.) Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y-variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question. A lawyer has found 60 investors for a limited partnership to purchase an inner-city apartment building, with each contributing either $5,000 or $10,000. If the partnership raised $405,000, then how many investors contributed $5,000 and how many contributed $10,000?

x =_________ $5,000 investors

y =_________ $10,000 investors

b.)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 $32,500 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 116,000 voters using the allocated funds?

Explanation / Answer

a) Let “X” number of investors invested $5,000 and “Y” number of investor invested $10,000 each.

Then, X + Y = 60………………….. Equation 1

So:

$5,000X + $10,000Y = $405,000
=> X + 2Y = ($405,000/$5,000)
=> X + 2Y = 81………………….. Equation 2

Subtract equation 1 from equation 2:

X + 2Y = 81
X + Y = 60
Y = 21

So, Y = 21 and
X = 60 – 21 = 39

Hence, number of investors contributing $5,000 is 39 and number of investors contributing $10,000 is 21.

b) Let say,

x=nos. of TV ads

y=nos. of Radio ads

Hence Following two equation can be formed:

3,000X + 500Y = 32,500
=> 6X + Y = 65……………. Equation 1

10,000X + 2,000Y = 116,000
=> 5X + Y = 58…….Equation 2

Subtract equation 2 from equation 1:

6X + Y = 65
5X + Y = 58
X = 7

Solve for Y by putting the value of X in equation 1:
6*7 + Y = 65
Y = 65 – 42 =23

Hence, number of TV ads would be 7 and number of Radio ads would be 23.

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