Question

Your purchase of CDs and DVDs has a total of 7 items. CDs cost $9 eachand DVDs cost $15 each. The total purchase costs $87. Write and solve asystem of linear equations to find the number r of CDs and the number yof DVDs that you purchased.

Answer 1

System of equations:

`\begin{gathered} x+y=7 \\ 9x+15y=87 \end{gathered}`

Solution:

`\begin{gathered} x=3 \\ y=4 \end{gathered}`

Step-by-step explanation:

Step 1. The given information is:

• There are 7 items between CDs and DVDs

,

• CDs cost $9

,

• DVDs cost $15

,

• The total of the purchase os $87

,

• x represents the number of CDs

,

• y represents the number of DVDs

Required: Write and solve a system of equations to find x and y.

Step 2. The first equation of the system will be an equation for the number of items. Since x is the number of CDs and y is the number of DVDs and there is a total of 7 items, the first equation is:

`x+y=7`

Step 3. The second equation of the system will be an equation for the value of the purchase. Since CDs cost $9, DVDs cost $15 and the total value of the purchase is $87, the second equation is:

`9x+15y=87`

Where 9x is the total value of the CDs, 15y is the total value of the DVDs and 87 is the result of adding these two values.

Step 4. the system of equations is:

`\begin{gathered} x+y=7 \\ 9x+15y=87 \end{gathered}`

Step 5. Now that we have the system of equations, we need to solve it in order to find x and y.

We will solve it using the elimination method, which consists of modifying the equations and then adding them to eliminate one of the variables.

The variable we will eliminate is x. For this, multiply the first equation of the system by -9:

`\begin{gathered} -9(x+y=7) \\ \downarrow\downarrow \\ -9x-9y=-63 \end{gathered}`

Step 6. Add the second equation of the system (shown in step 3 and 4) with the modified equation from step 5:

The x variable is eliminated, and we are left with an equation to find y:

`\begin{gathered} 6y=24 \\ \text{Solving for y:} \\ y=(24)/(6) \\ \boxed{y=4} \end{gathered}`

Step 7. Now that we know the value of y, we can find the value of x using the equation from step 2:

`\begin{gathered} x+y=7 \\ \text{Substituting y=4} \\ x+4=7 \end{gathered}`

Solving for x:

`\begin{gathered} x=7-4 \\ \boxed{x=3} \end{gathered}`

the value of x is 3, and the value of y is 4.

Answer:

System of equations:

`\begin{gathered} x+y=7 \\ 9x+15y=87 \end{gathered}`

Solution:

`\begin{gathered} x=3 \\ y=4 \end{gathered}`