# Marcus purchased 5 notebooks and 10 boxes of crayons for $31. Nina went to the same store and bought 10 notebooks and 5 boxes of crayons for$24.50. What is the cost of one notebook and one box of crayons?

Jan 16, 2018

$x = 1.20$
$y = 2.50$

#### Explanation:

$\text{Solving Process:}$
Let:
$x = \text{the price of the notebooks}$
$y = \text{the price of the boxes of crayons}$

Now, formulate equations with reference to their purchases; that is,
color(red)("Marcus ": 5x+10y=31->eq.1
color(blue)("Nina ": 10x+5y=24.50->eq.2

Then, solve the equations simultaneously as follows:

Multiply eq.1 with 2 to eliminate the terms with x variable in both equations.

eq.1-> color(red)(5x+10y=31) }-2
eq.2->color(blue)(10x+5y=24.5

$\text{so that the eq.1 becomes}$

eq.1->color(red)(cancel(-10x)-20y=-64
eq.2->color(blue)(cancel(10x)+5y=24.5

Then find the difference of the remaining terms to get the equation as shown below and find the value of $y$.

$\textcolor{red}{- 15 y = - 37.5}$; divide both sides by $- 15$ to isolate $y$
$\textcolor{red}{\frac{\cancel{- 15} y}{\cancel{- 15}} = \frac{- 37.5}{- 15}}$
color(red)(y=2.50; price for the boxes of crayons

Now, find the value of $x$, the price of the notebooks, by using either of the equations formulated. Here, eq.1 is used to solve for $x$.

$\textcolor{red}{5 x + 10 y = 31}$; where $\textcolor{red}{y = 2.50}$
$\textcolor{red}{5 x + 10 \left(2.50\right) = 31}$; simplify
$\textcolor{red}{5 x + 25 = 31}$; combine like terms
$\textcolor{red}{5 x = 31 - 25}$; simplify
$\textcolor{red}{5 x = 6}$; isolate $x$ by dividing both sides by $5$
$\textcolor{red}{x = 1.20}$; the price of the boxes of crayons

$\text{Checking Process} :$
where: $x = 1.20 \mathmr{and} y = 2.50$

$E q .1$
$5 x + 10 y = 31$
$5 \left(1.20\right) + 10 \left(2.50\right) = 31$
$6 + 25 = 31$
$31 = 31$

$E q .2$
$10 x + 5 y = 24.5$
$10 \left(1.20\right) + 5 \left(2.50\right) = 24.5$
$12 + 12.5 = 24.5$
$24.5 = 24.5$