Problem 12 - A Gentle Introduction to the Art of Mathematics

Question

Suppose that \(a, b, c\) and \(d\) are integers with \(a \neq c .\) Further, suppose that \(x\) is a real number satisfying the equation $$ \frac{a x+b}{c x+d}=1 $$ Show that \(x\) is rational. Where is the hypothesis \(a \neq c\) used?

Step-by-Step Solution

Answer

Given \( \frac{a x + b}{c x + d} = 1 \), solving gives \( x = \frac{d - b}{a - c} \). \(x\) is rational because it is a ratio of integers. The hypothesis \(a eq c\) ensures the denominator is not zero.

Step 1 - Set up the equation

Start with the given equation: \[ \frac{a x + b}{c x + d} = 1 \]

Step 2 - Clear the fraction

Multiply both sides of the equation by \(c x + d\) to clear the fraction: \[ a x + b = c x + d \]

Step 3 - Isolate the variable terms

Subtract \(c x\) from both sides to get terms involving \(x\) on one side: \[ a x - c x + b = d \]

Step 4 - Simplify

Combine the \(x\) terms on the left-hand side: \[ x (a - c) = d - b \]

Step 5 - Solve for \(x\)

Divide both sides by \(a - c\) to solve for \(x\): \[ x = \frac{d - b}{a - c} \]

Step 6 - Conclusion

Since \(a, b, c, d\) are integers, and the operations involve only integers and division, \(x\) is a ratio of integers. Therefore, \(x\) is a rational number.

Step 7 - Hypothesis usage

The condition \(a eq c\) ensures that the denominator \(a - c\) is not zero, which is essential for the fraction \(\frac{d - b}{a - c}\) to be defined.

Key Concepts

Equation Solving

Understand how to manipulate equations to find the value of a variable. We start with the given equation involving rational expressions. To simplify, clear the fraction by multiplying both sides by the denominator. This eliminates the fraction, leaving a linear equation.
After clearing the fraction, we isolate the variable terms. This involves moving all terms with the variable to one side. The other side will have the constant terms.
The next step is to simplify. Combine like terms to simplify the equation. Focus on gathering all variable terms together. Then solve for the variable, usually by isolating it.
In our exercise, these steps show that we can solve for the variable in most rational equations by isolating the terms and simplifying the equation. This logical progression is key to solving many algebraic problems.

Integer Properties

Integers are whole numbers, both positive and negative, including zero. They play a crucial role in this exercise. When we manipulate equations, the properties of integers ensure that operations such as addition, subtraction, and multiplication remain within the realm of integers.
The equation involves multiplying and adding integers. When we multiply them, the result is another integer. Likewise, when we add or subtract them, we still get an integer. This consistency is vital because it secures the integrity of our equation.
A key step is recognizing when to use division. We divide by the difference of two integers \(a - c\), ensuring it's not zero. This division results in a rational number, provided the numerator and denominator are integers. If we’ve followed all correct steps, and ensured that no division by zero occurs, we stay within the bounds of integers and rational numbers.

• Addition: \(a + b\) is always an integer if \(a\) and \(b\) are integers.
• Subtraction: \(a - b\) is always an integer if \(a\) and \(b\) are integers.
• Multiplication: \(a * b\) remains an integer if \(a\) and \(b\) are integers.
• Division: \(a / b\) is a rational number, provided \(b eq 0\).

These integer properties ensure that when combined with rational operations, the results are predictable and understandable.

Rationality in Mathematics

Rational numbers are any numbers that can be expressed as the quotient or fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q eq 0\). In mathematics, rational numbers provide a simple yet powerful means of understanding values and operations.
From our exercise, once we isolated \(x\), we expressed it as \(\frac{d - b}{a - c}\), a ratio of integers. This satisfies the definition of a rational number. Understanding the rationality concept helps in breaking down complex equations into simple components.
Rational relationships allow for predictable, repeatable outcomes. Being able to express a solution in the form of a ratio of integers gives mathematical clarity and precision. Knowing that \(x\) is rational provides confidence in the solution's validity.

• Rational numbers encompass whole numbers (e.g., \(5 = \frac{5}{1}\)).
• Fractions like \(\frac{3}{4}\) are rational because both numerator and denominator are integers.
• Rational operations (addition, subtraction, multiplication, and division) between rational numbers yield rational results.

Rationality is central to many mathematical disciplines, ensuring that values can be clearly defined and consistently worked with.