Problem 106 - Algebra and Trigonometry

Question

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 true only when \( a = 0 \); otherwise, it is false.

Step 1: Simplify the Right Side

The equation given is \( 2\left(\frac{a}{b}\right)=\frac{2a}{2b} \). We need to simplify the right side of the equation, \( \frac{2a}{2b} \). We can do this by dividing both the numerator and the denominator by 2, simplifying it to \( \frac{a}{b} \).

Step 2: Compare Both Sides

Now, after simplification, our equation becomes \( 2\left(\frac{a}{b}\right) = \frac{a}{b} \).

Step 3: Analyze Equality

The equation tells us that \( 2x = x \), where \( x = \frac{a}{b} \). This can only be true if \( x = 0 \), meaning that \( \frac{a}{b} = 0 \). For this equation to be true, \( a \) must be 0, regardless of the value of \( b \). However, for any other \( a eq 0 \), the equation does not hold.

Step 4: Conclusion

From Step 3, we conclude that the original equation \( 2\left(\frac{a}{b}\right)=\frac{2a}{2b} \) is not true for all values of \( a \) and \( b \); it is only true when \( a = 0 \) and \( b eq 0 \).

Key Concepts

Simplification

Simplification is a fundamental concept in solving equations. It involves reducing mathematical expressions to their simplest form. In the given equation, we started with \( 2\left(\frac{a}{b}\right)=\frac{2a}{2b} \). The right side, \( \frac{2a}{2b} \), can be simplified by dividing both the numerator and the denominator by 2. This process is crucial because it helps us see the equation more clearly and reveal relationships between variables.
The simplification gives us \( \frac{a}{b} \), making it much easier to handle compared to its previous form. By reducing the expression, we avoid potential complexity and the risk of miscalculations. Simplification is about making the math as straightforward as possible.

Numerator and Denominator

The concepts of numerator and denominator are key when dealing with fractions. A fraction \( \frac{a}{b} \) consists of a numerator, \(a\), which is the top part, and a denominator, \(b\), the bottom part. In our equation, we examined the fraction \( \frac{2a}{2b} \).
For simplification, notice both terms in the fraction are divisible by the same number, 2. This common factor allows you to reduce the equation to \( \frac{a}{b} \) by dividing both the numerator and the denominator by 2.
This notion is pivotal in ensuring fractions are represented in their simplest form and play a crucial role in comparing and simplifying complex expressions.

Analyzing Equality

Analyzing equality in equations is an essential step in understanding the relationship between variables. Our goal is to see whether the two sides of an equation are indeed equal for all values of variables. After reducing the original equation to \( 2\left(\frac{a}{b}\right) = \frac{a}{b} \), we needed to delve deeper.
To analyze this equation, observe that simplifying further tells us \( 2x = x \), letting \( x = \frac{a}{b} \). Such an equality holds true only under specific conditions due to the multiplication factor. Here, it shows that \( 2x \) can only equal \( x \) if \( x \) itself is zero. This implies the condition \( \frac{a}{b} = 0 \).
For this to hold, \( a \) must be zero, regardless of \( b \) as long as \( b \, eq \, 0 \). If \( a \) is not zero, the equality does not stand, illustrating how analyzing equality helps in determining the possible solutions or conditions of an equation.