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Magazine Chapter 5: Problem 15
\(\frac{x-5}{x+5}+\frac{x-7}{x+7}=2\)
Short Answer
Expert verified
The solution is \(x = -\frac{35}{6}\).
Step by step solution
01
Find a Common Denominator
To start solving the equation \( \frac{x-5}{x+5} + \frac{x-7}{x+7} = 2 \), we need a common denominator for the fractions. The denominators are \((x+5)\) and \((x+7)\). So the common denominator will be \((x+5)(x+7)\).
02
Rewrite Fractions with Common Denominator
Rewrite each fraction with the common denominator:\[ \frac{(x-5)(x+7)}{(x+5)(x+7)} + \frac{(x-7)(x+5)}{(x+5)(x+7)} = 2\]
03
Combine the Fractions
Combine the fractions into a single fraction:\[ \frac{(x-5)(x+7) + (x-7)(x+5)}{(x+5)(x+7)} = 2\]
04
Expand and Simplify Numerator
Expand each part of the numerator and simplify:- \((x-5)(x+7) = x^2 + 7x - 5x - 35 = x^2 + 2x - 35\)- \((x-7)(x+5) = x^2 + 5x - 7x - 35 = x^2 - 2x - 35\)Now combine:\[ x^2 + 2x - 35 + x^2 - 2x - 35 = 2x^2 - 70 \]
05
Solve the Equation
Substitute the expanded numerator into the equation:\[ \frac{2x^2 - 70}{(x+5)(x+7)} = 2 \]Cross-multiply to eliminate the fraction:\[ 2x^2 - 70 = 2(x+5)(x+7) \]
06
Expand the Right Side
Expand the right side of the equation:\( 2(x+5)(x+7) = 2(x^2 + 7x + 5x + 35) = 2x^2 + 24x + 70 \).
07
Equate and Simplify
Set the equation:\[ 2x^2 - 70 = 2x^2 + 24x + 70 \]Subtract \(2x^2\) from both sides:\[ -70 = 24x + 70 \]
08
Isolate x
Rearrange to solve for \(x\):Subtract 70 from both sides:\[ -70 - 70 = 24x \]Which simplifies to:\[ -140 = 24x \]Divide both sides by 24:\[ x = -\frac{35}{6} \]
09
Check the Solution
Substitute \(x = -\frac{35}{6}\) back into the original equation to verify it is correct. Upon calculation, both sides of the equation equal 2, confirming that \(x = -\frac{35}{6}\) is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
In any algebraic equation involving fractions, finding a common denominator is crucial. A common denominator enables us to add or subtract fractions with different denominators. In the exercise given, the fractions have denominators \((x+5)\) and \((x+7)\). To find a common denominator, we simply multiply these two expressions to get \((x+5)(x+7)\). This common denominator is essential because:
- It allows for the fractions to be combined easily.
- Facilitates further simplification of the equation.
Fraction Simplification
With algebra, simplifying fractions frequently involves expanding expressions and combining like terms. Once a common denominator has been established, the next step is to simplify the entire expression as much as possible. In the equation provided, the numerators \((x-5)(x+7)\) and \((x-7)(x+5)\) need to be expanded:
- For \((x-5)(x+7)\), expand to get \(x^2 + 2x - 35\)
- For \((x-7)(x+5)\), expand to get \(x^2 - 2x - 35\)
Cross-Multiplication
Cross-multiplication is a powerful tool used to eliminate fractions in equations, making them more straightforward to solve. In our exercise, after expressing the fractions under a common denominator and simplifying, we reach an equation:\[ \frac{2x^2 - 70}{(x+5)(x+7)} = 2 \]To clear the equation of fractions, cross-multiply. This involves multiplying each side by the denominator of the opposite fraction, effectively removing the fractions:
- Multiply the entire fraction by \((x+5)(x+7)\) to isolate the quadratic expression: \(2x^2 - 70 = 2(x+5)(x+7)\).
- This action allows the equation components to be expressed without fractions, simplifying the path to solving for \(x\).
