Question

in this problem, $a, b, c$, and $d$ are nonzero integers. if $\frac{a}{b}$ is added to $x$, the sum is $\frac{c}{d}$. which statement can be used to prove that $x$ must be a rational number?
$x = \frac{c - a}{d - b}$
$x = \frac{cb + ad}{bd}$
$x = \frac{cb - ad}{bd}$
$x = \frac{c + a}{d - b}$

Explanation:

Step1: Set up the given equation

We know $\frac{a}{b} + x = \frac{c}{d}$

Step2: Solve for $x$

Rearrange to isolate $x$: $x = \frac{c}{d} - \frac{a}{b}$

Step3: Find common denominator

Use common denominator $bd$: $x = \frac{cb}{bd} - \frac{ad}{bd}$

Step4: Combine the fractions

Subtract numerators: $x = \frac{cb - ad}{bd}$

Answer:

$x = \frac{cb - ad}{bd}$ (the third option)