Problem 42 - Algebra for College Students

Question

For Problems \(1-44\), solve each equation. $$ \frac{7}{x+4}=\frac{3}{x-8} $$

Step-by-Step Solution

Answer

The solution is \( x = 17 \).

Step 1: Cross-Multiply the Fractional Equation

To eliminate the fractions in the equation \( \frac{7}{x+4} = \frac{3}{x-8} \), cross-multiply to obtain an equation without fractions: \( 7(x-8) = 3(x+4) \). This gives us a more manageable linear equation to solve.

Step 2: Distribute and Simplify

Distribute the numbers on each side of the equation from Step 1. First, distribute the 7 on the left: \( 7 \times x - 7 \times 8 = 7x - 56 \). Then distribute the 3 on the right: \( 3 \times x + 3 \times 4 = 3x + 12 \). The equation becomes \( 7x - 56 = 3x + 12 \).

Step 3: Isolate the Variable Term

Move the variable terms to one side and the constant terms to the other. Subtract \( 3x \) from both sides to get \( 7x - 3x - 56 = 12 \). Simplifying this gives \( 4x - 56 = 12 \).

Step 4: Solve for the Variable

Add 56 to both sides of the equation to isolate the term with \( x \): \( 4x - 56 + 56 = 12 + 56 \). This simplifies to \( 4x = 68 \). Divide both sides by 4 to solve for \( x \): \( x = \frac{68}{4} = 17 \).

Step 5: Verify the Solution

To ensure the solution is correct, substitute \( x = 17 \) back into the original equation. Check both sides: \( \frac{7}{17+4} = \frac{7}{21} \) and \( \frac{3}{17-8} = \frac{3}{9} \), both simplify to \( \frac{1}{3} \). The solution \( x = 17 \) is verified.

Key Concepts

Cross-Multiplication

The first step in solving rational equations with fractions, like \( \frac{7}{x+4} = \frac{3}{x-8} \), often involves cross-multiplication. This method is useful for eliminating the fractions and simplifying the equation. Cross-multiplication means we multiply the numerator of each fraction by the denominator of the other fraction and set the products equal to each other. In our problem, we cross-multiply to obtain:

• Multiply \( 7 \times (x-8) \)
• Multiply \( 3 \times (x+4) \)

Doing so gives us the equation \( 7(x-8) = 3(x+4) \). This step transforms a rational equation into a linear one that's much easier to handle. Remember, always perform cross-multiplication carefully to avoid errors.

Linear Equations

After cross-multiplying, we derive a linear equation: \( 7(x-8) = 3(x+4) \). Linear equations have variables raised only to the first power, making them straightforward to solve. We can simplify this form using distribution, a process where we multiply through the parentheses:

• Distribute \( 7 \) to both \( x \) and \( -8 \) to get \( 7x - 56 \)
• Distribute \( 3 \) to both \( x \) and \( 4 \) to get \( 3x + 12 \)

Now, the equation reads \( 7x - 56 = 3x + 12 \). This representation is crucial for proceeding to variable isolation and solving the equation.

Variable Isolation

Variable isolation is the process of getting the variable by itself on one side of the equation. In our example, the equation simplifies to \( 7x - 56 = 3x + 12 \). We want to gather all terms involving \( x \) on one side. Follow these steps:

• Subtract \( 3x \) from both sides to get \( 7x - 3x - 56 = 12 \).
• This simplifies to \( 4x - 56 = 12 \).
• Next, add 56 to both sides to further simplify: \( 4x = 68 \).
• Finally, divide both sides by 4 to find \( x \): \( x = \frac{68}{4} = 17 \).

By isolating the variable, the solution \( x = 17 \) emerges, preparing it for verification.

Solution Verification

Solution verification ensures the solution satisfies the original equation. Substituting \( x = 17 \) back into \( \frac{7}{x+4} = \frac{3}{x-8} \) helps confirm its correctness:

• Left side: \( \frac{7}{21} = \frac{1}{3} \), since \( 21 = 17 + 4 \).
• Right side: \( \frac{3}{9} = \frac{1}{3} \), since \( 9 = 17 - 8 \).

Both sides simplify to the same value, verifying that \( x = 17 \) is correct. Verifying is a crucial step, especially with rational equations, to catch errors in calculation or logic, ensuring the solution truly works with the original problem.