Question

solve for $x$. $x = \square$ (type an integer or a simplified fraction. use a comma to separate answers as needed.)

Explanation:

Step1: Apply Triangle Proportionality Theorem

The theorem states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. So:
$$\frac{12}{7x} = \frac{4x+4}{10x-2}$$

Step2: Cross-multiply to eliminate fractions

Multiply the numerator of each ratio by the denominator of the other ratio:
$$12(10x-2) = 7x(4x+4)$$

Step3: Expand both sides of the equation

Calculate the products on each side:
$$120x - 24 = 28x^2 + 28x$$

Step4: Rearrange into standard quadratic form

Bring all terms to one side to set the equation to 0:
$$28x^2 + 28x - 120x + 24 = 0$$
$$28x^2 - 92x + 24 = 0$$

Step5: Simplify the quadratic equation

Divide all terms by 4 to reduce coefficients:
$$7x^2 - 23x + 6 = 0$$

Step6: Factor the quadratic equation

Find two numbers that multiply to $7\times6=42$ and add to -23, then factor:
$$(7x - 2)(x - 3) = 0$$

Step7: Solve for x using zero product property

Set each factor equal to 0 and solve:
$7x - 2 = 0 \implies x = \frac{2}{7}$
$x - 3 = 0 \implies x = 3$

Step8: Validate solutions

Check if solutions make denominators non-zero:
For $x=\frac{2}{7}$: $7x=2
eq0$, $10x-2=\frac{20}{7}-2=\frac{6}{7}
eq0$ (valid)
For $x=3$: $7x=21
eq0$, $10x-2=28
eq0$ (valid)

Answer:

$x = \frac{2}{7}, 3$