To solve the given proportion and derive the correct linear equation, let's go through a step-by-step process.
We are given the proportion:
[tex]\[
\frac{x - 9}{x - 3} = \frac{2}{5}
\][/tex]
To remove the fraction, we can cross-multiply. Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other.
So, we multiply:
[tex]\[
5 \times (x - 9) = 2 \times (x - 3)
\][/tex]
This gives us:
[tex]\[
5(x - 9) = 2(x - 3)
\][/tex]
Now, we distribute the numbers through the parentheses on both sides:
[tex]\[
5x - 45 = 2x - 6
\][/tex]
Our goal is to isolate [tex]\(x\)[/tex] on one side of the equation. To do that, we'll move all the terms involving [tex]\(x\)[/tex] to one side and the constant terms to the other side.
First, let's subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[
5x - 2x - 45 = -6
\][/tex]
[tex]\[
3x - 45 = -6
\][/tex]
Next, let's add 45 to both sides to isolate the term involving [tex]\(x\)[/tex]:
[tex]\[
3x - 45 + 45 = -6 + 45
\][/tex]
[tex]\[
3x = 39
\][/tex]
Finally, we solve for [tex]\(x\)[/tex] by dividing both sides by 3:
[tex]\[
x = \frac{39}{3}
\][/tex]
[tex]\[
x = 13
\][/tex]
So, the derived equation before simplifying further was:
[tex]\[
5x - 45 = 2x - 6
\][/tex]
Thus, the correct linear equation matching this derived form is:
[tex]\[
2x - 6 = 5x - 45
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{A} \: 2x - 6 = 5x - 45 \][/tex]
