Problem 16 - Algebra Form and Function

Question

Solve the equations. $$ \frac{2}{2-x}-\frac{3}{x-5}=0 $$

Step-by-Step Solution

Answer

Question: Solve the rational equation and find the value of x: $$\frac{2}{2-x} - \frac{3}{x-5} = 0$$. Answer: The solution for the given equation is $$x = \frac{16}{5}$$.

Step 1: Identify the common denominator.

The common denominator between \((2-x)\) and \((x-5)\) is the product of both \((2-x)(x-5)\).

Step 2: Clear the fractions.

Multiply both sides of the equation by the common denominator \((2-x)(x-5)\). This will eliminate the fractions: $$ (\frac{2}{2-x})(2-x)(x-5) - (\frac{3}{x-5})(2-x)(x-5) = 0\cdot(2-x)(x-5) $$

Step 3: Simplify the equation.

Simplify the equation by canceling out the common terms: $$ 2(x-5) - 3(2-x) = 0 $$

Step 4: Distribute and combine like terms.

Perform distribution and combine like terms: $$ 2x - 10 - 6 + 3x = 0 $$ $$ 5x - 16 = 0 $$

Step 5: Solve for x.

Now, we will isolate x by adding 16 to both sides and then dividing by 5: $$ 5x = 16 $$ $$ x = \frac{16}{5} $$

Step 6: Check the solution.

We will now plug the value of x back into the original equation, making sure that it doesn't make the denominator equal to zero: $$ \frac{2}{(2-\frac{16}{5})} - \frac{3}{(\frac{16}{5}-5)} = 0 $$ Since both denominators are non-zero and the equation holds true, the solution x is valid. So, the solution for the given equation is: $$ x = \frac{16}{5} $$

Key Concepts

Common Denominator

When dealing with equations containing fractions, one of the key steps is to find a common denominator. This is crucial because it helps to eliminate fractions and makes the equation easier to solve. Finding a common denominator involves identifying a number or expression that is a multiple of all the denominators in the equation. For instance, in the exercise with the fractions \( \frac{2}{2-x} \) and \( \frac{3}{x-5} \), the individual denominators are \((2-x)\) and \((x-5)\). To find the common denominator, we multiply the individual denominators together:

• \( (2-x)(x-5) \)

These are polynomial expressions, so we should consider their product when clearing fractions. Using this product as a multiplier across the entire equation allows us to work with a simpler, fraction-free equation. Understanding this technique is vital because it is the foundation for handling complex equations efficiently. It streamlines the problem-solving process, saving time and reducing mistakes.

Fractions in Equations

Fractions often appear in algebraic equations, adding a layer of complexity. However, understanding how to handle them can simplify the problem and lead to the solution much faster. Dealing with fractions in an equation typically involves either clearing them by multiplying through by the common denominator or working directly with them by addition, subtraction, etc.Here are some key points to remember:

• Identify all fractions and their denominators.
• Multiply the entire equation by the common denominator to remove the fractions.

In our specific exercise, we multiplied each term by \( (2-x)(x-5) \), and this allowed us to cancel the fractional terms, simplifying the equation to a more manageable form.This method ensures that we maintain the equality of the equation while transforming it into an easier form. It's essential to carefully choose the common denominators to avoid errors and to verify the solution by plugging it back into the original equation.

Solving Linear Equations

Once we have cleared the fractions from the equation, the next step is solving the resulting linear equation. A linear equation is an equation where the highest power of the variable is one. In our exercise, after clearing the fractions, the equation \( 2(x-5) - 3(2-x) = 0 \) simplifies further through distribution and combining like terms.Follow these key steps while solving linear equations:

• Distribute any coefficients over the terms inside parentheses.
• Combine like terms on each side of the equation.
• Isolate the variable on one side by using addition and subtraction.
• Finally, solve for the variable.

In our problem, distributing leads to \( 5x - 16 = 0 \). Adding 16 to both sides gives \( 5x = 16 \). Dividing both sides by 5, we find \( x = \frac{16}{5} \).These straightforward steps will help you solve almost any linear equation effectively. The final crucial step is to check your solution by substituting it back into the original equation to ensure it satisfies all conditions, such as not making any denominator zero.