A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio x_1 (in $) and the amount spent advertising in the newspaper x_2 (in $) according to y=ax_1+bx_2+c The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months. begin{array}{|c|c|c|}hline text { Advertising, } x_{1} & text { Advertising, } x_{2} &text{sales, y} hline $ 2400 & {$ 800} & {$ 36,000} hline $ 2000 & {$ 500} & {$ 30,000} hline $ 3000 & {$ 1000} & {$ 44,000} hlineend{array} a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model y=ax_1+bx_2+c d) Predict the monthly sales

A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio x_1 (in $) and the amount spent advertising in the newspaper x_2 (in $) according to y=ax_1+bx_2+c The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months. begin{array}{|c|c|c|}hline text { Advertising, } x_{1} & text { Advertising, } x_{2} &text{sales, y} hline $ 2400 & {$ 800} & {$ 36,000} hline $ 2000 & {$ 500} & {$ 30,000} hline $ 3000 & {$ 1000} & {$ 44,000} hlineend{array} a) Use the data to write a system of linear equations to solve for a, b, and c. b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix. c) Write the model y=ax_1+bx_2+c d) Predict the monthly sales

Question
Forms of linear equations
asked 2020-11-06
A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio \(x_1\)
(in $) and the amount spent advertising in the newspaper \(x_2\) (in $) according to \(y=ax_1+bx_2+c\)
The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months.
\(\begin{array}{|c|c|c|}\hline \text { Advertising, } x_{1} & \text { Advertising, } x_{2} &\text{sales, y} \\ \hline $ 2400 & {$ 800} & {$ 36,000} \\ \hline $ 2000 & {$ 500} & {$ 30,000} \\ \hline $ 3000 & {$ 1000} & {$ 44,000} \\ \hline\end{array}\)
a) Use the data to write a system of linear equations to solve for a, b, and c.
b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix.
c) Write the model \(y=ax_1+bx_2+c\)
d) Predict the monthly sales if the grocer spends $250 advertising on the radio and $500 advertising in the newspaper for a given month.

Answers (1)

2020-11-07
\(y=ax_1+bx_2+c\)
We have to determine the function:
\(\begin{cases}2400a+800b+c=36000\\ 2000a+500b+c=30000\\ 3000a+1000b+c=44000 \end{cases}\)
Set up a system of equations for a,b,c:
\(\begin{bmatrix}2400&800&1&|&36000\\2000&500&1&|&30000\\3000&1000&1&|&44000\end{bmatrix}\)
b) Build the augmented matrix:
\(\begin{bmatrix}1&0&0&|&12\\0&1&0&|&4\\0&0&1&|&4000\end{bmatrix}\)
Use a graphing utility to find the reduced row-echelon form of the augmented matrix: \(a=12 , b=4 , c=4000\)
c) Determine a,b,c:
\(y=12x_1+4x_2+4000\)
write the model y that fits the data:
\(y=12(2500)+4(500)+4000=36000\)
d) Determine y for \(x_1=2500 , x_2=500\)
0

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