Cost of 1 shirt is [tex]\$25[/tex] and cost of one sweater is [tex]\$26[/tex]
Solution:
Given that Jane’s buying clothes.
She can get 5 shirts & 4 sweaters for [tex]\$229[/tex]
She can get 2 shirts & 3 sweaters for [tex]\$128[/tex]
Need to determine cost of 1 shirt and 1 sweater.
Let assume cost of 1 shirt = [tex]\$x[/tex]
And assume cost of 1 sweater = [tex]\$y[/tex]
Let us create equations from give conditions
First condition is Cost of 5 shirt + cost of 4 sweater = [tex]\$229[/tex]
[tex]\begin{array}{l}{\text { cost of 5 shirts }=5 \times \text { cost of 1 shirt }=5 \times x=5 x} \\\\ {\text { cost of 4 sweater }=4 \times \text { cost of 1 sweater }=4 \times y=4 y}\end{array}[/tex]
So using first condition equation which we get is as follows
[tex]5 x+4 y=229 \rightarrow (1)[/tex]
Second condition is Cost of 5 shirt + cost of 4 sweater = [tex]\$128[/tex]
[tex]\begin{array}{c}{\text { cost of } 2 \text { shirts }=2 \times \text { cost of } 1 \text { shirt }=2 \times x=2 x} \\\\ {\text { cost of } 3 \text { sweater }=3 \times \text { cost of } 1 \text { sweater }=3 \times y=3 y}\end{array}[/tex]
So using second condition equation which we get is as follows
[tex]2 \mathrm{x}+3 \mathrm{y}=128\quad\rightarrow (2)[/tex]
Now we have two equations to be solved
[tex]\begin{array}{l}{5 x+4 y=229 \rightarrow (1)} \\\\ {2 x+3 y=128 \rightarrow (2)}\end{array}[/tex]
On multiplying equation (1) by 2 and equation (2) by 5 to make coefficients of x equal in both equations, we get
[tex]\begin{array}{l}{2 x(5 x+4 y)=2\times 229} \\\\ {10 x+8 y=458 \rightarrow (3)} \\\\ {5 x(2 x+3 y)=5 \times 128} \\\\ {10 x+15 y=640 \rightarrow (4)}\end{array}[/tex]
On subtracting (3) from (4), we get
[tex]\begin{array}{l}{(10 \mathrm{x}-10 \mathrm{x})+(15 \mathrm{y}-8 \mathrm{y})=640-458} \\\\ {\Rightarrow 7 \mathrm{y}=182} \\\\ {\Rightarrow \mathrm{y}=\frac{182}{7}=26}\end{array}[/tex]
On substituting y = 26 in equation (1) we get
[tex]\begin{array}{l}{\Rightarrow 5 \mathrm{x}+(4 \times 26)=229} \\\\ {\Rightarrow 5 \mathrm{x}=229-104} \\\\ {\Rightarrow 5 \mathrm{x}=125} \\\\ {\Rightarrow \mathrm{x}=\frac{125}{5}=25}\end{array}[/tex]
Cost of 1 shirt = [tex]\$x = \$25[/tex]
Cost of 1 sweater = [tex]\$y = \$26[/tex]
