Problem 62 - Algebra 1

Question

What is the solution of the equation \(\frac{9}{x+5}=\frac{7}{x-5} ?\) (A) 5 (B) 8 (C) 20 (D) 40 (E) 80

Step-by-Step Solution

Answer

The solution for x is 40, hence answer choice (D) is correct.

Step 1: Identify the extremes and means

First, identify the extremes (a and d) and means (b and c) in the proportion. From the given equation, \( \frac{9}{x+5}=\frac{7}{x-5}\), the extremes are 9 and \( x - 5 \) while the means are \( x+5 \) and 7.

Step 2: Apply cross-multiplication

In a proportion a/b=c/d, the product of the means equals the product of the extremes. That is, a x d = c x b. Apply this here: 9 x (x - 5) = 7 x (x + 5). This will solve to 9x - 45 = 7x + 35.

Step 3: Simplify the equation

Simplify the equation by substracting 7x from both sides to get 2x - 45 = 35. Then, add 45 to both sides to get 2x = 80.

Step 4: Solve for x

Finally, divide both sides by 2 to solve for x. The solution is x = 40.

Key Concepts

Cross-multiplication

Cross-multiplication is a savvy method used to solve equations in the form of proportions. When you see an equation like \( \frac{9}{x+5}=\frac{7}{x-5} \), you're looking at a proportion. Here, cross-multiplication allows you to simplify the task by breaking it down into a straightforward calculation.

To cross-multiply, follow these simple steps:

• Recognize both sides as fractions set equal to each other.
• Multiply one numerator by the opposite denominator. In our example, multiply 9 by \( x-5 \).
• Do the same for the other pair. Multiply 7 by \( x+5 \).
• Set the results from these multiplications equal to each other.

In equation form, this looks like: \( 9 \times (x - 5) = 7 \times (x + 5) \).

This key step transforms your problem into a simpler linear equation, which is easier to solve. Cross-multiplication is particularly useful for clearing fractions in proportional equations.

Proportion

A proportion is an equation stating that two ratios are equivalent. In the expression \( \frac{9}{x+5}=\frac{7}{x-5} \), a proportion shows that the ratio of 9 to \( x+5 \) is the same as 7 to \( x-5 \). Understanding proportions is crucial in math as it sets the stage for using cross-multiplication.

Here are quick points to grasp proportions:

• A ratio compares two numbers or quantities.
• A proportion gives a statement of equality between two such ratios.
• The proportions imply that when one part of a ratio changes, the other part changes at the same rate.

In solving proportions, if adjusting one side, remember to do the equivalent operation to the other to maintain equality. This consistency is what allows cross-multiplication to work neatly.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange an equation or expression in a particular way. After cross-multiplication provides the equation \( 9x - 45 = 7x + 35 \), algebraic manipulation is what helps you solve for \( x \).

Steps to perform effective algebraic manipulation:

• Simplify: Begin by getting all terms involving \( x \) on one side by subtracting \( 7x \) from both sides. This renders \( 2x - 45 = 35 \).
• Isolate the variable: Next, remove any constant from \( x \) by adding 45 to both sides, leading to \( 2x = 80 \).
• Solve: Finally, divide both sides by 2 to get \( x = 40 \).

Algebraic manipulation is a seamless way to strip down equations to their core elements, aiding in finding the unknown.