Question

Shelly has $\mathbf{\backslash (}\mathbf{175}$to shop for jeans and sweaters. Each pair of jeans cost $\mathbf{\backslash )}\mathbf{25}$, each sweater costs $\mathbf{\$}\mathbf{20}$, and she buys 8 items. Determine the number of pairs of jeans and sweaters Shelly bought.

Short answer

Shelly bought 3 pairs of jeans and 5 sweaters.

Step-by-step solution

Step 1. State the concept of Linear System with two variables.

A linear system of two equations with two variables is any system that can be written in the form.

$\begin{array}{l}ax+by=p\\ cx+dy=q\end{array}$

Where any of the constants can be zero with the exception that each equation must have at least one variable in it.

Also, the system is called linear if the variables are only to the first power, are only in the numerator and there are no products of variables in any of the equations.

Step 2. Write the equations from the conditions given in the question.

Assume the number of pair of jeans purchased by Shelly to be$x$.

(Note:$A$pair of jeans means 1 pair of jeans.$A$pair means two joined corresponding parts that aren't used separately. In other words, a pair of jeans is 1 pair of jeans.)

Assume the number of sweaters purchased by Shelly to belocalid="1647669016276" $y$.

Each pair of jeans cost$\$25$.

$\begin{array}{c}\text{Thereforecostof\hspace{0.17em}\hspace{0.17em}}x\text{\hspace{0.17em}\hspace{0.17em}pairsofjeans}=25\times x\\ =\$25x\end{array}$

Each sweater costslocalid="1647669033679" $\$20$.

$\begin{array}{c}\text{Thereforecostof\hspace{0.17em}\hspace{0.17em}}y\text{\hspace{0.17em}\hspace{0.17em}sweaters=20}\times \text{y}\\ \text{=\$20y}\end{array}$

From the$1^{st}$condition,

Shelly has$\$175$in total to make the purchase.

Therefore the first equation is given as follows:

$25x+20y=175\dots \left(1\right)$

From the $2nd$condition,

Shelly buys in total of 8 items.

$x+y=8\dots \left(2\right)$

Step 3. Solve the above system of equations by substitution method.

Substitution method: Thesubstitution method is the algebraic method to solve simultaneous linear equations. As the word says, in this method, the value of one variable from one equation is substituted in the other equation. In this way, a pair of the linear equation gets transformed into one linear equation with only one variable, which can then easily be solved.

The equation (2) can be written as:

$\begin{array}{l}x+y=8\\ y=8-x\dots \left(3\right)\end{array}$

Substitute $y=8-x$in equation (1) and simplify further to find the value of $‘x’$.

$\begin{array}{l}25x+20y=175\\ 25x+20(8-x)=175\left[\text{from\hspace{0.17em}\hspace{0.17em}equation\hspace{0.17em}\hspace{0.17em}}\left(3\right)\right]\\ 25x+160-20x=175\\ 25x-20x=175-160\\ 5x=15\\ x=\frac{15}{5}\\ x=3\end{array}$

Substitute the value $x=3$ in equation (2) and simplify further to find the value of ‘y’.

$\begin{array}{c}x+y=8\\ 3+y=8\\ y=8-3\\ y=5\end{array}$

Therefore, Shelly purchased 3 pairs of jeans and 5 sweaters.