Problem 57 - Algebra 2

Question

Solve each equation. Check your solution. $$ \frac{5}{2-x}=\frac{4}{2 x+1} $$

Step-by-Step Solution

Answer

The solution to the given equation is \(x=\frac{3}{14}\).

Step 1: Cross-multiply

Start by cross-multiplying to get rid of the fraction on both sides. For this equation, this would look like: \(5*(2x+1) = 4*(2-x)\).

Step 2: Expand and Simplify

Expand and simplify the equation. By doing so, we get: \(10x+5 = 8-4x\).

Step 3: Move all x-terms to one side

Move all terms with \(x\) to one side and the constants to the other. This will result in: \(10x + 4x = 8 - 5\). Simplifying this will give: \(14x = 3\).

Step 4: Solve for x

Finally, solve for \(x\) by dividing both sides by 14. The solution is: \(x=\frac{3}{14}\).

Step 5: Check the solution

Now, substitute \(x = \frac{3}{14}\) back into the original equation to check that the solutions obtained don't make any denominator zero, and it satisfies the original equation as well.

Key Concepts

Understanding Cross-Multiplication

Cross-multiplication is a technique used to solve rational equations, where you eliminate the fractions by multiplying across the equal sign.
It simplifies the process and makes the equation easier to solve. To use cross-multiplication:

• Multiply the numerator of one fraction by the denominator of the other.
• Do the same for the other fraction.

For example, in the equation \( \frac{5}{2-x} = \frac{4}{2x+1} \):

• Multiply 5 by \((2x+1)\), and 4 by \((2-x)\).

This results in the equation: \(5(2x+1) = 4(2-x)\), eliminating the fractions altogether.

The Importance of Checking Solutions

Checking your solution is crucial to ensure accuracy, especially in rational equations.
It's important because:

• Substituting back ensures that the solution satisfies the original equation.
• It confirms that the denominator never becomes zero, as this would make the expression undefined.

For example, substitute \(x = \frac{3}{14}\) back into the original equation:

• Check each side separately to see if they are equal.

If both sides match, your solution is correct.

Mastering Algebraic Manipulation

Algebraic manipulation is the skill of rearranging equations to isolate variables, simplifying complicated expressions, and solving for unknowns.
Here's a breakdown:

• First, expand the terms by distributing the numbers outside parentheses.
• Next, simplify by combining like terms.
• Rearrange terms to isolate the variable on one side.

In our example:

• Expand: \(10x + 5 = 8 - 4x\).
• Rearrange: \(10x + 4x = 8 - 5\), which simplifies to \(14x = 3\).

This process unveils the solution step by step.

Exploring Rational Expressions

Rational expressions involve ratios of polynomials, similar to fractions, and can pose unique challenges.
They require understanding both numerators and denominators:

• Numerators and denominators are polynomials.
• Simplifying requires factoring and cancelling terms whenever possible.

For equations like \( \frac{5}{2-x} = \frac{4}{2x+1} \), ensure:

• Common factors are simplified correctly.
• Any restrictions, such as values causing the denominator to be zero, are considered.

Mastering rational expressions lays a strong foundation for tackling complex equations.