and d are nonzero integers…

Maths Questions

Step by Step Solution

Mathematics Questions

If a, b, c, and d are nonzero integers, and adding ( frac{b}{a} ) to x results in ( frac{d}{c} ), which statement can be used to prove that x must be a rational number? A. ( x = frac{b}{d} – frac{c}{b} – frac{a}{d} ) B. ( x = frac{b}{d} + frac{c}{b} + frac{a}{d} ) C. ( x = d – frac{b}{c} – a ) D. ( x = d – frac{b}{c} + a )

Short Answer

This proof demonstrates that if a and b are nonzero integers, we can manipulate the equation a/b + x = c/d to isolate x as x = (cb – ad) / bd. Since both the numerator and denominator are products of integers, x is confirmed to be a rational number.

Step-by-Step Solution

Step 1: Understanding the Variables

In this proof, we are working with nonzero integers: a, b, c, and d. These integers will be integral to our calculations. Specifically, we start with the ratio a/b which represents a rational number. The objective is to manipulate this expression to solve for x, while keeping our focus on maintaining rationality throughout the process.

Step 2: Setting Up the Equation

The equation starts with adding the rational number a/b to an unknown x, leading us to the equation: a/b + x = c/d. We want to isolate x to demonstrate that it is also a rational number. To do this, we will subtract a/b from both sides of the equation, enabling us to express x in terms of c/d and a/b.

Step 3: Isolating x and Deriving the Final Equation

After rearranging the equation, we arrive at: x = c/d – a/b. To combine these fractions, we need a common denominator, which is bd. Thus, we manipulate the equation to find: x = (cb – ad) / bd. As both the numerator (cb – ad) and the denominator (bd) are products of integers, we conclude that x must indeed be a rational number.

Related Concepts

Nonzero integers

Integers that are not equal to zero, which can be either positive or negative

Rational number

A number that can be expressed as the quotient or fraction of two integers, where the denominator is not zero

Common denominator

A shared multiple of the denominators of two or more fractions, used to add or subtract those fractions effectively.

Table Of Contents

If a, b, c, and d are nonzero integers, and adding ( frac{b}{a} ) to x results in ( frac{d}{c} ), which statement can be used to prove that x must be a rational number? A. ( x = frac{b}{d} – frac{c}{b} – frac{a}{d} ) B. ( x = frac{b}{d} + frac{c}{b} + frac{a}{d} ) C. ( x = d – frac{b}{c} – a ) D. ( x = d – frac{b}{c} + a )