Question

The cost of 5 notebooks and 3 pens is $\mathrm{\backslash (}9.75$. The cost of 4 notebooks and 6 pens is $\mathrm{\backslash )}10.50$. Which of the following systems can be used to find the cost of a notebook $n$and a pen $p$?

a. Write a system of equations to model the situation.

b. Solve the system of equations. How much does each item cost?

Short answer

a. The system of equations to model the situation is:

$\begin{array}{c}5n+3p=9.75\\ 4n+6p=10.50\end{array}$

b. The cost of each notebook and pen are $\$1.5$ and $\$0.75$respectively.

Step-by-step solution

Part a. Step 1. Write the equations for the given problem.

Let the cost of 1 notebook be$n$ and the cost of 1 pen be $p$.

It is given that cost of 5 notebooks and 3 pens is $\$9.75$.

Therefore, the equation depicting the above information is:$5n+3p=9.75$

It is given that the cost of 4 notebooks and 6 pens is $\$10.50$.

Therefore, the equation depicting the above information is:

$4n+6p=10.50$

Part a. Step 2. Write a system of equations to model the situation.

Therefore, the system of equations to model the situation are:

$\begin{array}{c}5n+3p=9.75\\ 4n+6p=10.50\end{array}$

Part b. Step 1. Number the equations in the system of equations.

The equations after numbering is:

$\begin{array}{c}5n+3p=9.75\left(1\right)\\ 4n+6p=10.50\left(2\right)\end{array}$

Multiply the both sides of the equation (1) by 2.

$\begin{array}{c}2\left(5n+3p\right)=2\left(9.75\right)\\ 10n+6p=19.50\left(3\right)\end{array}$

Subtract the equation (2) from the equation (3).

Therefore, it is obtained that:

role="math" localid="1647688345770" $$\begin{gathered} 10n+6p-\left(4n+6p\right)=19.50-10.50 \\ 10n+6p-4n-6p=9 \\ 6n=9 \\ n=\frac{9}{6} \\ n=\frac{3}{2} \\ n=1.5 \end{gathered}$$

Therefore, the value of$n$is 1.5.

Substitute the value of $n$in equation (1).

$$\begin{gathered} 5n+3p=9.75 \\ 5\left(1.5\right)+3p=9.75 \\ 7.5+3p=9.75 \\ 3p=9.75-7.5 \\ 3p=2.25 \\ p=\frac{2.25}{3} \\ p=0.75 \end{gathered}$$

Therefore, the value of$p$ is 0.75.

Therefore, the cost of each notebook and pen$\$1.5$ are$\$0.75$ and respectively.