Question

Two neighbors bought plants from the same store to improved their yards. James spend $77 on 7 daylilies and 8 shrubs. Liz spent $140 on 14 daylilies and 14 shrubs. what is the cost of one Daylily and one shrub?

Answer 1

Step 1. First, let's define the variable we are going to use:

`\begin{gathered} d\longrightarrow\text{ price of a daylily} \\ s\longrightarrow\text{price of a shrub} \end{gathered}`

Step 2. Find the first equation from the information about James.

James spends $77 on 7 daylilies and 8 shrubs, this is described in the following equation:

`7d+8s=77`

Step 3. To find the second equation we use the information about Liz.

Liz spends $140 on 14 daylilies and 14 shrubs. The equation describing this is:

`14d+14s=140`

Since all of the numbers involved in this equation are multiples of 14, we can divide the whole equation by 14 to simplify it:

`\begin{gathered} (14d+14s=140)/(14) \\ \\ d+s=10 \end{gathered}`

Step 4. Define the system of equations.

From step 2 and step 3 we have our two equations:

`\begin{gathered} 7d+8s=77 \\ d+s=10 \end{gathered}`

Step 5. The method we will use to solve this system of equations is the substitution method. To use this method, we solve one of the equations for 1 variable. In our case, we solve the second equation for d:

`\begin{gathered} d+s=10 \\ d=10-s \end{gathered}`

And the second step of the substitution method is to substitute that value into the first equation:

`\begin{gathered} 7d+8s=77 \\ \text{Substituting d=10-s} \\ 7(10-s)+8s=77 \end{gathered}`

Step 6. Solve the previous equation for s.

`7(10-s)+8s=77`

using the distributive property to multiply 7 by 10 and 7 by -s:

`70-7s+8s=77`

Combining like terms:

`70+s=77`

Subtracting 70 to both sides of the equation:

`\begin{gathered} s=77-70 \\ s=7 \end{gathered}`

The cost of one shrub is $7.

Step 7. Now that we know the value of s, we can find the value of d using the equation we had previously solved for d in step 5:

`d=10-s`

Substituting s=7:

`\begin{gathered} d=10-7 \\ d=3 \end{gathered}`

The cost of one daylily is $3.

Answer:

The cost of one shrub is $7 and the cost of one daylily is $3.