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Sagot :
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]
[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]
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