Problem 107 - 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.) $$ \frac{-a}{b}=-\frac{a}{b} $$

Step-by-Step Solution

Answer

The equation is true for all values of \( a \) and \( b \) except \( b = 0 \).

Step 1: Understand the Equation

The given equation is \( \frac{-a}{b} = -\frac{a}{b} \). We need to determine if this is true for all values of the variables \(a\) and \(b\). Disregard any values that make the denominator zero, meaning \(b eq 0\).

Step 2: Simplify the Equation

Rearrange and simplify the expression on the right-hand side to determine equality. The right side of the equation \(-\frac{a}{b}\) is equivalent to \(\frac{-a}{b}\).

Step 3: Compare the Two Sides

Compare both sides of the equation: \(\frac{-a}{b}\) and \(\frac{-a}{b}\). Notice they are identical, confirming the initial equation is consistent.

Step 4: Conclusion

Since both sides of the equation \(\frac{-a}{b} = \frac{-a}{b}\) are identical, the equation is indeed true for all values of \(a\) and \(b\) (except when \(b = 0\)).

Key Concepts

Equation Simplification

Simplifying rational expressions involves making each side of the equation look as simple as possible without changing their value. In the equation \( \frac{-a}{b} = -\frac{a}{b} \), both expressions can undergo simplification. When you simplify, you're essentially showing that the expressions have the same value.To simplify the right-side expression, \( -\frac{a}{b} \), you multiply both the numerator and the expression by \(-1\). This leads back to \( \frac{-a}{b} \). This simplified form helps us easily see that both expressions on either side of the equation are equal.

• Understand what you're simplifying: Sometimes expressions look different but mean the same.
• Aim for clarity: Make the equation easier to read and compare.
• Keep the values equivalent: The numerical value must stay constant in simplification.

This demonstrates how simplification is a powerful tool for verifying equality in math.

Variable Restriction

Variable restriction is critical when working with rational expressions, especially when they appear in the denominator. In our equation \( \frac{-a}{b} = -\frac{a}{b} \), the variable \( b \) is in the denominator.Whenever dealing with denominators, remember:

• Avoid zero: If \( b = 0 \), the expression is undefined because division by zero is not permissible in mathematics.
• Identify restricted values: In this exercise, \( b eq 0 \).
• Assume other values are valid: As long as the denominator is not zero, the equation holds true.

By acknowledging variable restrictions, we ensure our solution is mathematically consistent and accurately represents all permissible values.

Equality of Expressions

Determining the equality of expressions means verifying that two different-looking mathematical phrases represent the same value for all permissible variable values. In \( \frac{-a}{b} = -\frac{a}{b} \), we analyze if both sides equate.Once simplified, both the left and right sides are \( \frac{-a}{b} \). Thus, each side is identical in value and appearance, which confirms the equation is true for every value of \( a \) and any non-zero \( b \).

• Check each side: Ensure individual expressions behave the same way under all scenarios.
• Use simplification: Making expressions simpler aids in quickly spotting equality.
• Confirm consistency: The original equation will hold true.

Recognizing equal expressions ensures that solutions are valid across all possible cases, except where operations are undefined.