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{cb - ad}{bd}\\) \\(x = \frac{c + a}{d - b}\\) \\(x = \frac{c - a}{d - b}\\) \\(x = \frac{cb + ad}{bd}\\)

Explanation:

Step1: Set up the given equation

The problem states $\frac{a}{b} + x = \frac{c}{d}$.

Step2: Solve for $x$

Subtract $\frac{a}{b}$ from both sides:
$x = \frac{c}{d} - \frac{a}{b}$

Step3: Find common denominator

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

Step4: Verify rationality

Since $a,b,c,d$ are nonzero integers, $cb-ad$ and $bd$ are integers, and $bd
eq 0$, so $x$ is rational.

Answer:

$x = \frac{cb - ad}{bd}$