Question

Eight stethoscopes and three endoscopes cost $378.02, whereas three stethoscopes and eight endoscopes cost $513.97. How much does onestethoscope cost?

Answer 1

Let's call x the cost of one stethoscope and y the cost of each endoscope, then:

$\begin{gathered} 8x+3y=378.02\text{ Eq(1)} \\ 3x+8y=513.97\text{ Eq(2)} \end{gathered}$

We have a system of linear equations. To find the x-value and y-value, start by solving for x in equation 1:

$\begin{gathered} 8x=378.02-3y \\ x=\frac{378.02-3y}{8}\text{ Eq(3)} \end{gathered}$

Now, replace this into Equation 2:

$\begin{gathered} 3\cdot(\frac{378.02-3y}{8})+8y=513.97 \\ \frac{3\cdot(378.02-3y)}{8}+8y=513.97 \\ \text{Apply the distributive property} \\ \frac{3\cdot378.02-3\cdot3y}{8}+8y=513.97 \\ \frac{1134.06-9y}{8}+8y=513.97 \\ \text{Apply the properties of fractions} \\ \frac{1134.06}{8}-\frac{9y}{8}+8y=513.97 \\ 141.758-1.125y+8y=513.97 \\ -1.125y+8y=513.97-141.758 \\ 6.875y=372.212 \\ y=\frac{372.212}{6.875} \\ y=54.14 \end{gathered}$

Now, replace the y-value into equation 3, and find x:

$\begin{gathered} x=\frac{378.02-3(54.14)}{8} \\ x=\frac{378.02-162.42}{8} \\ x=\frac{215.6}{8} \\ x=26.95 \end{gathered}$

Then, one stethoscope costs $26.95.