To derive the linear equation from the given proportion:
[tex]\[ \frac{x-9}{x-3}=\frac{2}{5} \][/tex]
we will follow these steps:
1. Cross-Multiply: We eliminate the fraction by cross-multiplying the terms. This means we multiply the numerator of the left-hand side by the denominator of the right-hand side, and multiply the denominator of the left-hand side by the numerator of the right-hand side:
[tex]\[ 5 \cdot (x - 9) = 2 \cdot (x - 3) \][/tex]
2. Distribute the Constants: We then distribute the constants (5 and 2) on both sides of the equation:
[tex]\[ 5(x - 9) = 2(x - 3) \][/tex]
[tex]\[ 5x - 45 = 2x - 6 \][/tex]
3. Rearrange the Equation: Next, we want to collect all the x terms on one side of the equation and the constant terms on the other side. First, we can subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ 5x - 2x - 45 = -6 \][/tex]
[tex]\[ 3x - 45 = -6 \][/tex]
4. Isolate x: We add 45 to both sides to isolate the term with x:
[tex]\[ 3x - 45 + 45 = -6 + 45 \][/tex]
[tex]\[ 3x = 39 \][/tex]
5. Solve for x: Finally, we divide both sides by 3 to solve for x:
[tex]\[ x = 13 \][/tex]
However, the problem asks which linear equation can be derived from the given proportion, not the value of [tex]\(x\)[/tex]. Hence, we should identify correctly how our final simplified form matches up with the options provided.
From our cross-multiplying and simplification, we get the linear equation:
[tex]\[ 5(x - 9) = 2(x - 3) \][/tex]
or equivalently,
[tex]\[ 5x - 45 = 2x - 6 \][/tex]
Upon comparing with the answer choices, we see that:
- Option D states [tex]\(2 x - 6 = 5 x - 45\)[/tex], which is the same as our derived equation since it only rearranges the terms.
Therefore, the correct linear equation derived from given proportion is:
[tex]\[ \boxed{2 x - 6 = 5 x - 45} \][/tex]
