Problem 102 - 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.) $$ \frac{b}{b-c}=1-\frac{b}{c} $$

Step-by-Step Solution

Answer

The equation is not true for all values of variables satisfying \( b \neq c \) and \( c \neq 0 \).

Step 1: Identify Domain Restrictions

Observe that the equation \( \frac{b}{b-c} = 1 - \frac{b}{c} \) involves division, meaning the denominators cannot be zero. Therefore, we have two restrictions: \( b-c eq 0 \) and \( c eq 0 \). Thus, \( b eq c \) and \( c eq 0 \).

Step 2: Simplify Right Side of the Equation

The right side of the equation is \( 1 - \frac{b}{c} \). Rewrite 1 as \( \frac{c}{c} \), thus:\[ 1 - \frac{b}{c} = \frac{c}{c} - \frac{b}{c} = \frac{c-b}{c} \]

Step 3: Equate and Cross-Multiply

Equate the fractions from both sides of the equation:\[ \frac{b}{b-c} = \frac{c-b}{c} \]Cross-multiply to eliminate fractions:\[ b \cdot c = (c-b) \cdot (b-c) \]

Step 4: Expand and Simplify

Expand the right-hand side of the equation:\( (c-b) \cdot (b-c) = cb - bc - b^2 + c^2 \).The equation thus becomes:\[ bc = cb - bc - b^2 + c^2 \]

Step 5: Rearrange and Check for Equality

By rearranging, we simplify the equation:\[ bc = cb - bc - b^2 + c^2 \]\( 2bc = cb - b^2 + c^2 \)\( 2bc = bc - b^2 + c^2 \)We observe anomalies here; since the equations no longer match, indicating a lack of equivalence.

Key Concepts

Domain Restrictions

When dealing with algebraic equations that include division, it's crucial to consider domain restrictions. Domain restrictions arise in order to prevent division by zero, which is undefined in mathematics. In the equation \( \frac{b}{b-c} = 1 - \frac{b}{c} \), there are denominators \( b-c \) and \( c \).
To find the restrictions, ask yourself: what values of the variables would make the denominator zero?

For instance:

• \( b - c eq 0 \): This means that \( b \) cannot equal \( c \).
• \( c eq 0 \): The variable \( c \) itself cannot be zero, as it stands alone in the denominator.

Therefore, the domain for this equation is restricted to all values of \( b \) and \( c \) except the ones mentioned in these restrictions.

Fraction Division

Understanding fraction division is key to manipulating equations with fractions. In our equation, look at the way fractions are present on both sides. This reminds us how division of one fraction can be re-cast into a multiplication problem.
Consider the expression \( 1 - \frac{b}{c} \) from our equation, which we view as a subtraction involving fractional division:

• The fraction \( \frac{b}{c} \) acts as a divisor in the context of that subtraction.

Fraction division is embedded in many algebra problems, thus knowing how to handle fractional terms, like converting expressions and simplifying, helps in solving equations efficiently.

Equation Simplification

Simplifying equations is a tactic to reduce complexity in algebraic expressions. Take our equation to see how simplification plays out: \( 1 - \frac{b}{c} \) becomes \( \frac{c}{c} - \frac{b}{c} \).
What happens here transforms a mixed number into a simple fraction:

• Replace 1 with \( \frac{c}{c} \) to unify the denominators, which are now the same, allowing easy subtraction.
• This subtracts directly, yielding \( \frac{c-b}{c} \), a crucial step in simplification.

This method showcases how reshaping expressions can clarify and simplify algebraic operations.

Cross-Multiplication

Cross-multiplication is a technique used to solve equations involving fractions. It eliminates the denominators by transforming the equation through multiplication.
Given the equal fractions \( \frac{b}{b-c} = \frac{c-b}{c} \), cross-multiplication steps are:

• Multiply diagonally: \( b \cdot c \) and \( (c-b) \cdot (b-c) \).
• This action results in \( b \cdot c = c \cdot (b-c) - b \cdot (b-c) \), expanding further reveals complex yet manageable terms.

Cross-multiplication offers a straightforward method to clear out fractions and solve the equation, showing how powerful and simplifying this technique can be.