# Claire bought three bars of soap and five sponges for $2.31. Steve bought five bars of soap and three sponges for$3.05. How do you find the cost of each item?

Jan 12, 2017

A bar of soap costs 52¢, or $0.52 A sponge costs 15¢, or $0.15

#### Explanation:

This question involves setting up a system of equations. If you are not familiar with systems of equations, I'd suggest you watch this video before proceeding.

Now, to set up a system, we need variables. Let's call the price of one soap bar $x$, and the price of one sponge $y$. With this information, we can construct the following:

Claire:
3 soaps and 5 sponges for $2.31 $\implies 3 x + 5 y = 231$Steve: 5 soaps and 3 sponges for$3.05 $\implies 5 x + 3 y = 305$

Note: I have converted dollars to cents, simply to keep everything in whole numbers. We'll convert back to dollars in the end.

Now we have a system. Notice, however, that nothing cancels by simply adding or subtracting equations. The common step is to manipulate one equation so that things will cancel, but doing this here will lead to a lot of messy fractions. Hence, we will manipulate both equations. We will multiply Claire's equation by 5, and Steve's equation by 3. This gives us:

$15 x + 25 y = 1155$
$15 x + 9 y = 915$

Now, notice that we have a $15 x$ in both equations. This means that if we subtract one equation from another, the $x$'s will cancel each other out completely, leaving us with one variable - $y$ - to solve for. As shown:

$\left(15 x + 25 y = 1155\right) - \left(15 x + 9 y = 915\right)$

$\implies 16 y = 240$

Now, to solve for $y$. Dividing both sides by 16 give us:

$y = \frac{240}{16} = 15$

Now that we know what $y$ is, we can plug it into any of our two initial equations, and solve for $x$. I will chose Claire's equation (the first one):

$3 x + 5 \left(15\right) = 231$
$\implies 3 x = 156$
$\implies x = 52$

Now we have $x$ and $y$, let's go back to what they actually mean in the context of this problem:

A bar of soap ($x$) costs 52¢, or $0.52 A sponge ($y$) costs 15¢, or $0.15

Hope that helped :)