To determine the value of [tex]\( x \)[/tex] that makes [tex]\( \overline{ KM } \parallel \overline{ JN } \)[/tex] using the given proportion
[tex]\[
\frac{x-5}{x} = \frac{x-3}{x+4},
\][/tex]
we will solve this step-by-step.
### Step-by-Step Solution
1. Given Equation:
[tex]\[
\frac{x-5}{x} = \frac{x-3}{x+4}
\][/tex]
2. Cross-multiply to clear the fractions:
[tex]\[
(x-5)(x+4) = x(x-3)
\][/tex]
3. Distribute the terms:
- Distribute on the left side:
[tex]\[
x(x) + x(4) - 5(x) - 5(4) = x^2 + 4x - 5x - 20
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 - x - 20
\][/tex]
- On the right side, distribute [tex]\( x \)[/tex] through [tex]\( (x-3) \)[/tex]:
[tex]\[
x(x) - x(3) = x^2 - 3x
\][/tex]
4. Set the expressions equal to each other:
[tex]\[
x^2 - x - 20 = x^2 - 3x
\][/tex]
5. Subtract [tex]\( x^2 \)[/tex] from both sides:
[tex]\[
-x - 20 = -3x
\][/tex]
6. Isolate [tex]\( x \)[/tex]:
[tex]\[
-x + 3x = 20
\][/tex]
Simplify this to:
[tex]\[
2x = 20
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 10
\][/tex]
So, the value of [tex]\( x \)[/tex] that makes [tex]\( \overline{ KM } \parallel \overline{ JN } \)[/tex] is
[tex]\[
\boxed{10}
\][/tex]
