Problem 11 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ \frac{p}{p-2}=\frac{2}{5} $$

Step-by-Step Solution

Answer

Solution: \( p = \frac{-4}{3} \).

Step 1: Cross-Multiply

To solve for \( p \), first apply the cross-multiplication method to the equation \( \frac{p}{p-2} = \frac{2}{5} \). Cross-multiplying gives us \( 5p = 2(p-2) \).

Step 2: Distribute and Simplify

Distribute the 2 on the right side of the equation: \( 2(p-2) = 2p - 4 \), resulting in \( 5p = 2p - 4 \). Simplify this to get rid of the \( 2p \) on the right by subtracting \( 2p \) from both sides: \( 5p - 2p = -4 \).

Step 3: Solve for p

Simplifying the left side gives us \( 3p = -4 \). Solve for \( p \) by dividing both sides by 3: \( p = \frac{-4}{3} \).

Step 4: Check the Solution

Substitute \( p = \frac{-4}{3} \) back into the original equation to verify it's a valid solution: \( \frac{-4/3}{(-4/3)-2} \). Simplify the denominator: \( \frac{-4/3}{-4/3 - 6/3} \). This becomes \( \frac{-4/3}{-10/3} = \frac{2}{5} \), which matches the original right side, confirming that \( \frac{-4}{3} \) is indeed a solution.

Key Concepts

Equation Solving

Solving equations is a fundamental process in algebra where we find the value of a variable that satisfies a given equation. The equation provided is a rational equation, involving fractions. To solve it, we need to isolate the variable, which in this case is \( p \). This often involves steps such as:

• Rearranging terms to simplify both sides of the equation.
• Using operations like addition, subtraction, multiplication, and division to both sides of the equation to maintain equality.
• Checking the solution by substituting it back into the original equation to ensure that both sides are equal.

This approach emphasizes logic and strategic manipulation of expressions to find valid solutions.

Rational Equations

Rational equations are equations that involve ratios of polynomials. These types of equations can be tricky, but a common strategy to solve them is using cross-multiplication. Cross-multiplication allows us to eliminate the fractions, transforming the equation into a simpler form. Given our example, we have:\[ \frac{p}{p-2} = \frac{2}{5} \]Applying cross-multiplication, we multiply across the equals sign to get rid of the fractions:\[ 5p = 2(p-2) \] This results in an equation without any seesaw effects of fractions, making it easier to deal with. Just remember, it’s crucial to check your solutions in rational equations, as sometimes they can lead to false results (extraneous solutions) where the original equation could have undefined sections, like division by zero.

Fraction Simplification

Simplification is key when dealing with fractions, especially in solving rational equations. It makes the problem easier to manage and the calculations more straightforward. The goal is to ensure the fractions are in their simplest form. Here are the steps you might take for simplification:

• Identify common factors in the numerator and the denominator and divide them out.
• Convert complex fractions to simpler forms when needed, ensuring accurate calculations.

For example, confirming the solution \( \frac{-4/3}{-10/3} = \frac{2}{5} \) requires simplifying the fraction by dividing numerators and denominators by common terms. Fraction simplification allows us to focus on the broader picture of the equation, thus validating our solutions more smoothly and efficiently.