To determine the correct linear equation derived from the proportion:
[tex]\[
\frac{12}{x+8} = \frac{7}{x+1}
\][/tex]
we will follow a methodical approach step by step:
1. Cross Multiplication:
We start by cross-multiplying to eliminate the denominators:
[tex]\[
12 \cdot (x + 1) = 7 \cdot (x + 8)
\][/tex]
2. Distribution:
Distribute the constants 12 and 7 across the expressions in parentheses:
[tex]\[
12(x + 1) = 7(x + 8)
\][/tex]
[tex]\[
12x + 12 = 7x + 56
\][/tex]
3. Rearranging the Equation:
To form the linear equation, we need to gather all the [tex]\(x\)[/tex]-terms on one side of the equation and the constant terms on the other side. Subtract [tex]\(7x\)[/tex] from both sides:
[tex]\[
12x + 12 - 7x = 7x + 56 - 7x
\][/tex]
[tex]\[
5x + 12 = 56
\][/tex]
Now, subtract 12 from both sides to isolate the [tex]\(x\)[/tex]-term:
[tex]\[
5x = 56 - 12
\][/tex]
[tex]\[
5x = 44
\][/tex]
Divide by 5 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{44}{5}
\][/tex]
Although solving for [tex]\(x\)[/tex] might be helpful, here we need to identify the derived linear equation from the proportion. Notice that the format given in the problem asks about the original step after applying cross multiplication but before we isolate [tex]\(x\)[/tex]. Therefore, comparing the equations we generated after distribution gives us:
[tex]\[
12x + 12 = 7x + 56
\][/tex]
Thus, the correct linear equation is:
[tex]\[
\boxed{12x + 12 = 7x + 56}
\][/tex]
So the correct answer is:
C. [tex]\(12 x + 12 = 7 x + 56\)[/tex]
