Problem 16 - Algebra Form and Function

Question

Solve the equations. $$ \frac{9}{x-3}-\frac{5}{1-x}=0 $$

Step-by-Step Solution

Answer

Answer: There is no solution for x in this equation.

Step 1: Identify the common denominator

Since the denominators are (x-3) and (1-x), we can notice that they are related by a negative sign. So, they can be treated as the same denominator. The common denominator will be (x-3).

Step 2: Rewrite the equation with the common denominator

Our equation should look like: $$ \frac{9}{x-3}-\frac{5}{(x-3)(-1)}=0 $$ Which simplifies to $$ \frac{9}{x-3}+\frac{5}{x-3}=0 $$

Step 3: Add the fractions and solve for x

Now we can add the fractions to simplify the equation: $$ \frac{9+5}{x-3}=0 $$ $$ \frac{14}{x-3}=0 $$ In this step, for a fraction to equal zero, the numerator must be zero. The numerator in this case is 14, which is not zero. So, we can conclude that there is no solution for x for this equation.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions where the numerators and/or denominators contain algebraic expressions. These equations are distinguished by having variables in the denominator. Understanding rational equations is crucial, as they appear frequently in algebra and calculus.

When dealing with rational equations, it is important to first identify if the equation is valid throughout its domain. Since dividing by zero is undefined, certain values for the variable may make the denominator zero, leading to restrictions on solutions.

• Check if expressions in the denominators could equal zero.
• Identify the excluded values that make denominators zero.
• Remember to state these restrictions after solving the equation.

Learning to solve rational equations will help you in tackling more complex algebraic problems effectively.

Common Denominator

To solve equations involving fractions, establishing a common denominator is a key step. A common denominator lets you combine fractions and simplifies the equation. Here’s how you can find and use a common denominator in rational equations:

Start by looking at the denominators in your fractions. Finding a common denominator is essential because it allows you to combine fractions into a single fraction, making the equation easier to work with.

• Identify the least common multiple (LCM) of all denominators.
• Rewrite each fraction with this common denominator.
• Perform addition or subtraction of the fractions.

In our example, the denominators \((x-3)\) and \((1-x)\) are essentially the same, but with opposite signs. This illustrates the importance of recognizing equivalent expressions. Rewriting the equation with the same denominator allows for the easy addition of fractions, leading toward a simplified solution.

Solving Equations

Solving algebraic equations consists of finding values for variables that satisfy the equation. With rational equations, once you establish a common denominator, your next goal is to simplify and solve.

Once the equation is rewritten with a common denominator, as it was in the example, you add or subtract any numerators. In the example provided, combining terms into a single fraction yields: \[ \frac{14}{x-3} = 0 \]At this stage, check the numerator of the fraction. Remember, a fraction equals zero only if its numerator is zero, provided the denominator is not zero. Here, since the numerator is non-zero (14 in this case), the equation resolves to having no solution. This is sometimes surprising but important to note:

• Combining fractions transforms the equation.
• Zero in the numerator implies the whole fraction equals zero.
• The absence of a zero in the numerator means the equation doesn't have a valid solution in its domain.

By understanding each step, from forming common denominators to evaluating equations, solving becomes a straightforward process.