Question

Esmeralda is buying some new shirts and sweaters. She is able to buy 5 shirts and 4 sweaters for $267 or she is able to buy 3 shirts and 5 sweaters for $259. How much does a shirt cost? How much does a sweater cost?

Answer 1

Let the cost of a shirt be x, and let the cost of a sweater be y.

The cost of 5 shirts will be 5x and the cost of 4 sweaters will be 4y.

Since it is given that 5 shirts and 4 sweaters cost $267, it follows that:

`5x+4y=267`

This gives the first equation.

The cost of 3 shirts will be 3x and the cost of 5 sweaters will be 5y.

Since it is given that 3 shirts and 5 sweaters cost $259, it also follows that:

`3x+5y=259`

This forms the second equation.

Next, solve equations simultaneously:

`\begin{gathered} 5x+4y=267 \\ 3x+5y=259 \\ \text{Multiply the first equation by 5 and the second equation by 4:} \\ 25x+20y=1335 \\ 12x+20y=1036 \\ \text{Subtract the second equation from the first to get:} \\ 25x-12x+20y-20y=1335-1036 \\ \Rightarrow13x+0=299 \\ \Rightarrow13x=299\Rightarrow(13x)/(13)=(299)/(13) \\ \Rightarrow x=23 \end{gathered}`

Substitute the x value, x=23 into the first equation to get the value of y.

`\begin{gathered} 5x+4y=267 \\ \Rightarrow5(23)+4y=267\Rightarrow115+4y=267 \\ \Rightarrow4y=267-115\Rightarrow4y=152 \\ \Rightarrow(4y)/(4)=(152)/(4)\Rightarrow y=38 \end{gathered}`

Hence, the cost of a shirt is $23 and the cost of a sweater is $38.