Respuesta :
Step 1. First, let's define the variable we are going to use:
[tex]\begin{gathered} d\longrightarrow\text{ price of a daylily} \\ s\longrightarrow\text{price of a shrub} \end{gathered}[/tex]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:
[tex]7d+8s=77[/tex]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:
[tex]14d+14s=140[/tex]Since all of the numbers involved in this equation are multiples of 14, we can divide the whole equation by 14 to simplify it:
[tex]\begin{gathered} \frac{14d+14s=140}{14} \\ \\ d+s=10 \end{gathered}[/tex]Step 4. Define the system of equations.
From step 2 and step 3 we have our two equations:
[tex]\begin{gathered} 7d+8s=77 \\ d+s=10 \end{gathered}[/tex]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:
[tex]\begin{gathered} d+s=10 \\ d=10-s \end{gathered}[/tex]And the second step of the substitution method is to substitute that value into the first equation:
[tex]\begin{gathered} 7d+8s=77 \\ \text{Substituting d=10-s} \\ 7(10-s)+8s=77 \end{gathered}[/tex]Step 6. Solve the previous equation for s.
[tex]7(10-s)+8s=77[/tex]using the distributive property to multiply 7 by 10 and 7 by -s:
[tex]70-7s+8s=77[/tex]Combining like terms:
[tex]70+s=77[/tex]Subtracting 70 to both sides of the equation:
[tex]\begin{gathered} s=77-70 \\ s=7 \end{gathered}[/tex]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:
[tex]d=10-s[/tex]Substituting s=7:
[tex]\begin{gathered} d=10-7 \\ d=3 \end{gathered}[/tex]The cost of one daylily is $3.
Answer:
The cost of one shrub is $7 and the cost of one daylily is $3.
