Problem 21 - Algebra for College Students

Question

For Problems \(1-44\), solve each equation. $$ \frac{-2}{x-5}=\frac{1}{x+9} $$

Step-by-Step Solution

Answer

The solution for the equation is \(x = \frac{-13}{3}\).

Step 1: Identify the Equation and Clear the Fractions

The given equation is \( \frac{-2}{x-5}=\frac{1}{x+9} \). To solve the equation with fractions, the first step is to cross-multiply to eliminate the fractions. This means multiplying each side by the denominator of the other fraction. This gives us: \[-2(x+9) = 1(x-5)\].

Step 2: Distribute and Simplify

Next, we'll distribute the terms on both sides of the equation:\(-2(x+9)\) becomes \(-2x - 18\), and\(1(x-5)\) becomes \(x - 5\). The equation is now: \[-2x - 18 = x - 5\].

Step 3: Combine Like Terms

To isolate \(x\), move all terms involving \(x\) to one side of the equation and constant terms to the other side. Add \(2x\) to both sides: \(-18 = 3x - 5\) Next, add \(5\) to both sides to isolate the \(x\) terms: \(-18 + 5 = 3x\), which simplifies to \(-13 = 3x\).

Step 4: Solve for x

Divide both sides by \(3\) to solve for \(x\): \[x = \frac{-13}{3}\].

Step 5: Final Verification

Substitute \(x = \frac{-13}{3}\) back into the original equation to ensure both sides are equal. Calculate: The left-hand side is \(\frac{-2}{\frac{-13}{3} - 5}\) and the right-hand side is \(\frac{1}{\frac{-13}{3} + 9}\). Both sides simplify to the same value, verifying the solution is correct.

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to solve equations that feature fractions. By cross-multiplying, you can eliminate the denominators, making the equation easier to solve. Consider the equation \( \frac{-2}{x-5}=\frac{1}{x+9} \). Here are the steps involved in cross-multiplication:

• Multiply the numerator of the left fraction by the denominator of the right fraction.
• Multiply the numerator of the right fraction by the denominator of the left fraction.

For our example, this results in: \(-2(x+9) = 1(x-5)\). By doing this, we rewrite the equation without fractions, simplifying our path to finding the solution. Cross-multiplication is valuable because it transforms a fraction equation into a simple linear equation.

Distributive Property

The distributive property helps to expand expressions that involve multiplication across addition or subtraction inside parentheses. For the given equation \(-2(x+9) = 1(x-5)\), we apply the distributive property to both sides. By applying the distributive property:

• \(-2(x+9)\) becomes \(-2x - 18\)
• \(1(x-5)\) becomes \(x - 5\)

These steps simplify the equation to \(-2x - 18 = x - 5\). The distributive property is essential in simplifying complex expressions and is widely used in algebra to help eliminate parentheses, providing a clearer path to isolating the variable.

Isolating Variables

Isolating variables involves rearranging an equation so that the unknown variable appears on one side, making it easier to solve the equation. Starting with our simplified equation \(-2x - 18 = x - 5\), the objective is to solve for \(x\). To isolate \(x\), follow these steps:

• Add \(2x\) to both sides to get \(-18 = 3x - 5\)
• Next, add \(5\) to both sides to obtain \(-13 = 3x\)
• Finally, divide both sides by \(3\) to solve for \(x\), resulting in \(x = \frac{-13}{3}\)

Isolating the variable simplifies the process of finding its value and is a fundamental principle in algebra.

Equation Verification

Equation verification is the process of checking whether the solution obtained actually satisfies the original equation. After isolating \(x\) and calculating \(x = \frac{-13}{3}\), it is critical to plug this back into the original equation to confirm the solution.The original equation is \( \frac{-2}{x-5}=\frac{1}{x+9} \). Replace \(x\) with \(\frac{-13}{3}\):

• The left side simplifies as \(\frac{-2}{\frac{-13}{3} - 5}\)
• The right side simplifies as \(\frac{1}{\frac{-13}{3} + 9}\)

After simplifying both sides, they should equal the same value, confirming the solution \(x = \frac{-13}{3}\) is correct. Equation verification ensures accuracy and reliability in finding solutions to algebraic problems.