Problem 94 - Algebra and Trigonometry

Question

\(89-96\) m State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ 2\left(\frac{a}{b}\right)=\frac{2 a}{2 b} $$

Step-by-Step Solution

Answer

The equation is not true for all values; it's only true when \(a = 0\).

Step 1: Simplify Left Side of Equation

The left side of the equation is given as \(2\left(\frac{a}{b}\right)\). To simplify, distribute the 2 across the fraction: \[2 \times \frac{a}{b} = \frac{2a}{b}.\]

Step 2: Simplify Right Side of Equation

The right side of the equation is \(\frac{2a}{2b}\). Notice that the numerator and the denominator have a common factor of 2. Simplify this fraction by dividing both the numerator and the denominator by 2: \[\frac{2a}{2b} = \frac{a}{b}.\]

Step 3: Compare the Simplified Expressions

Now we compare the two simplified sides. The left side is \(\frac{2a}{b}\) and the right side is \(\frac{a}{b}\). These two expressions are not equal unless \(a = 0\). Therefore, the equation is not true for all values of \(a\) and \(b\).

Key Concepts

Simplifying Fractions

Fractions might seem tricky, but they become much easier if we simplify them. Simplifying fractions means reducing them to their simplest form. This involves dividing both the numerator (top number) and the denominator (bottom number) by their greatest common factor. Let's see this in action: in the example given, the right side of the equation is \( \frac{2a}{2b} \). Both the numerator and the denominator share a common factor of 2. Thus, we can simplify it by dividing 2 from each, resulting in \( \frac{a}{b} \). Simplifying fractions is a crucial step in solving and comparing equations, as it often reveals whether different expressions might actually represent the same value. If fractions are not simplified, you might incorrectly conclude that two expressions are different.

Common Factors

Common factors are numbers or variables that exactly divide the components of an expression without leaving any remainder. They play a pivotal role in simplification. When simplifying fractions, identifying these common factors allows us to reduce the fraction to its simplest form. For instance, in the fraction \( \frac{2a}{2b} \), 2 is a common factor for both the numerator and the denominator. By dividing both by 2, the expression becomes \( \frac{a}{b} \), simplifying our problem significantly. Consistently spotting and using common factors helps in many algebraic processes, reducing complexity and making equations easier to solve. Without identifying common factors, we might miss simplifying expressions correctly.

Variable Values

Variables, often represented by letters such as \( a \) or \( b \), stand in for unknown numbers in equations. Understanding what values these variables can take is important for solving and verifying equations. In our example, we needed to determine whether \( 2 \left( \frac{a}{b} \right) = \frac{2a}{2b} \) holds true for all variable values. After simplification, it becomes evident that \( \frac{2a}{b} eq \frac{a}{b} \) unless \( a = 0 \). This analysis shows us that the equation is not universally true, as substituting most values into \( a \) won't yield equality. Understanding how variable values affect an equation helps prevent mathematical errors, especially in verification tasks like these.

Denominator Restrictions

Denominators, the numbers or variables beneath the fraction line, have restrictions to prevent making the fraction undefined. One of the most critical restrictions is that a denominator cannot be zero, as division by zero is undefined in mathematics. In the provided exercise, any value of \( b \) that equals zero would make expressions like \( \frac{a}{b} \) or \( \frac{2a}{b} \) undefined. Therefore, while verifying equations, it's crucial to disregard values that violate this rule. Ensuring we account for these restrictions helps in accurately determining the truth of the equation for the possible range of variable values without running into undefined scenarios.