To derive a linear equation from the given proportion:
[tex]\[
\frac{12}{x+8} = \frac{7}{x+1}
\][/tex]
follow these steps:
1. Cross-Multiply: To remove the fractions, we'll cross-multiply the terms of the proportion. This involves multiplying the numerator of each fraction by the denominator of the other fraction.
[tex]\[
12 \cdot (x + 1) = 7 \cdot (x + 8)
\][/tex]
2. Distribute: Apply the distributive property to both sides of the equation to expand the expressions.
[tex]\[
12(x + 1) = 7(x + 8)
\][/tex]
Expanding both sides, we get:
[tex]\[
12x + 12 = 7x + 56
\][/tex]
3. Combine Like Terms: To isolate [tex]\( x \)[/tex] on one side, we need to get all [tex]\( x \)[/tex] terms on one side and the constant terms on the other side.
First, subtract [tex]\( 7x \)[/tex] from both sides:
[tex]\[
12x - 7x + 12 = 56
\][/tex]
Simplifying this, we get:
[tex]\[
5x + 12 = 56
\][/tex]
Next, subtract 12 from both sides:
[tex]\[
5x = 44
\][/tex]
Finally, divide both sides by 5:
[tex]\[
x = \frac{44}{5}
\][/tex]
However, our task is not to find [tex]\( x \)[/tex], but to determine the correct linear equation resulting from the proportion.
The expanded form before simplifying was:
[tex]\[
12x + 12 = 7x + 56
\][/tex]
Therefore, the correct linear equation derived from the given proportion is:
[tex]\[
12x + 12 = 7x + 56
\][/tex]
Thus, the correct answer is:
B. [tex]\( 12x + 12 = 7x + 56 \)[/tex]
