Let's solve the given equation step by step to determine at what number of months the costs of the two companies are equal, and to find the simplified version of the equation.
The equation given is:
[tex]\[ 9 + 12m = 6 + 15(m - 1) \][/tex]
Step 1: Distribute the 15 on the right side
The right-hand side of the equation expands as follows:
[tex]\[ 15(m - 1) = 15m - 15 \][/tex]
So, the equation now is:
[tex]\[ 9 + 12m = 6 + 15m - 15 \][/tex]
Step 2: Combine like terms
Combine the constant terms on the right side:
[tex]\[ 6 - 15 = -9 \][/tex]
Thus, we have:
[tex]\[ 9 + 12m = 15m - 9 \][/tex]
Step 3: Subtract 12m from both sides to get all terms with [tex]\(m\)[/tex] on one side
This gives:
[tex]\[ 9 = 3m - 9 \][/tex]
Step 4: Add 9 to both sides to isolate the term with [tex]\(m\)[/tex]
This gives:
[tex]\[ 18 = 3m \][/tex]
Step 5: Divide both sides by 3 to solve for [tex]\(m\)[/tex]
This gives:
[tex]\[ m = 6 \][/tex]
At [tex]\(6\)[/tex] months, the prices of the two companies are the same.
Now, let’s identify the simplified version of the equation from the options given:
1. [tex]\(3m = 15\)[/tex]
2. [tex]\(9 + 15m = 12m - 1\)[/tex]
3. [tex]\(3m = 18\)[/tex]
4. [tex]\(9 + 6 - 15 = 15m - 12m\)[/tex]
The correct simplified version we derived is:
[tex]\[ 3m = 18 \][/tex]
Therefore, the choices are:
COMPLETE: At [tex]\(\boxed{6}\)[/tex] months, the prices are the same.
Thus, the simplified equation is [tex]\(3m = 18\)[/tex].
