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Magazine Chapter 4: Problem 42
For Problems \(1-44\), solve each equation. $$ \frac{7}{x+4}=\frac{3}{x-8} $$
Short Answer
Expert verified
The solution is \( x = 17 \).
Step by step solution
01
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.
02
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 \( 7x - 56 = 3x + 12 \).
03
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 \( 4x - 56 = 12 \).
04
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 \( x \): \( x = \frac{68}{4} = 17 \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( 3 \times (x+4) \)
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 \( 3x + 12 \)
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 \( x \): \( x = \frac{68}{4} = 17 \).
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 \( 9 = 17 - 8 \).
