Problem 11 - Algebra and Trigonometry

Question

Determine whether the given value is a solution of the equation. \(\frac{X-a}{X-b}=\frac{a}{b} \quad(b \neq 0)\) (a) \(x=0 \quad\) (b) \(x=b\)

Step-by-Step Solution

Answer

The value \(x = 0\) is a solution; \(x = b\) is not a solution (undefined).

Step 1: Understand the Equation

The equation given is \( \frac{X-a}{X-b}=\frac{a}{b} \) where \(b eq 0\). Our goal is to determine if a particular value for \(x\) satisfies this equation.

Step 2: Substitute x = 0 into the Equation

First, we will check if \(x = 0\) is a solution. Substitute \(X = 0\) into the equation: \( \frac{0-a}{0-b} = \frac{a}{b} \). Simplifying the left side gives \( \frac{-a}{-b} = \frac{a}{b} \). This simplifies to \( \frac{a}{b} = \frac{a}{b} \), which is true. So, \(x = 0\) is a solution.

Step 3: Substitute x = b into the Equation

Next, check if \(x = b\) is a solution. Substitute \(X = b\) into the equation: \( \frac{b-a}{b-b} = \frac{a}{b} \). The left side becomes \( \frac{b-a}{0} \), which is undefined. Therefore, \(x = b\) cannot be a solution as the equation is not defined for this value.

Key Concepts

Rational Equations

Rational equations are mathematical expressions where one fraction, often involving polynomials, is set equal to another fraction. These types of equations are common in algebra problems and have specific techniques for solving them. In this exercise, the rational equation is depicted as \( \frac{X-a}{X-b}=\frac{a}{b} \). The aim here is to find out if certain values for \(x\) satisfy or solve this equation.

When working with rational equations, it's important to remember the following:

• Simplify both sides of the equation if possible.
• Clear the fractions by finding a common denominator if needed.
• Check for any restrictions, such as values that make the denominator zero, which are undefined.

By following these steps, solving rational equations becomes a manageable process that can reveal valid solutions or expose undefined expressions.

Substitution Method

The substitution method is a useful technique in algebra for determining if a specific value is a solution to an equation. The process involves substituting the potential solution into the equation and simplifying to see if both sides remain equal.

Here's how to apply the substitution method step by step:

• Take the given value for \(x\) and substitute it into the rational equation.
• Simplify each side of the equation after substitution.
• Check if the simplified expressions on both sides of the equation are equal. If they are, the value is a solution.

In the original exercise, substituting \(x = 0\) into the rational equation confirmed it as a solution since both sides simplified to the same value. However, when substituting \(x = b\), the equation became undefined due to a zero denominator, which brings us to the next important topic.

Undefined Expressions

Undefined expressions occur in mathematics when operations are attempted that do not yield valid numerical results. A common cause of undefined expressions in rational equations is division by zero.

When you substitute a value into a rational equation, it's crucial to check if the denominator becomes zero, as this would make the expression undefined.

To avoid undefined expressions:

• Identify the values that make any denominator zero before substituting.
• Remember that any such value cannot be a solution to the equation.

In the problem we discussed, substituting \(x = b\) led to an undefined situation because it resulted in division by zero in the denominator \(X - b\). This clearly shows that not all substitutions will result in a valid solution, highlighting the importance of checking for undefined expressions in rational equations.