Problem 4 - Beginning and Intermediate Algebra

Question

Solve the following proportions. \(\frac{15}{8}=\frac{x}{x-63}\)

Step-by-Step Solution

Answer

The solution to the proportion \(\frac{15}{8} = \frac{x}{x - 63}\) is \(x = 135\).

Step 1: 1. Cross-multiplication

In order to solve the proportion, we'll cross-multiply: \[15(x - 63) = 8x\]

Step 2: 2. Expanding the expression

Now, we'll expand the expression on the left side of the equation: \[15x - 945 = 8x\]

Step 3: 3. Simplifying the equation

We will simplify the equation by isolating the \(x\) terms on one side by subtracting 8x from both sides: \[7x - 945 = 0\]

Step 4: 4. Solving for x

Finally, we'll solve for \(x\) by adding 945 to both sides and dividing by 7: \[7x = 945 \Rightarrow x = \frac{945}{7}\] So, \(x = 135\).

Key Concepts

Cross-Multiplication

Cross-multiplication is a useful method for solving proportions. It involves multiplying the numerator of one fraction by the denominator of the other fraction. In our problem, we deal with the proportion \(\frac{15}{8}=\frac{x}{x-63}\). Here, we multiply 15 by \((x - 63)\) and 8 by \(x\). This gives us the equation:

• \(15(x - 63) = 8x\)

Cross-multiplication helps us to eliminate the fractions, making it easier to deal with algebraic equations. This method simplifies the process and is a powerful tool to have in your algebra toolkit. Remember, only use this technique when you're dealing with two ratios set equal to each other, as this is when proportions are defined.

Solving for x

Once you have cross-multiplied a proportion, the next step is tackling the algebraic equation to find the value of \(x\). From the equation

• \(15(x - 63) = 8x\)

we continue by expanding it. First, distribute 15 over the terms in the parenthesis:

• \(15x - 945 = 8x\)

We need to isolate \(x\) on one side of the equation. By subtracting \(8x\) from both sides, we get:

• \(7x - 945 = 0\)

Finally, to solve for \(x\), we add 945 to both sides:

• \(7x = 945\)
• Divide both sides by 7, resulting in \(x = 135\)

This process shows how algebraic manipulation helps us find the unknown variable.

Algebraic Equations

Algebraic equations are mathematical statements that have constants and variables connected by operations like addition or multiplication. In solving proportional problems, you'll often form equations where you need to solve for a variable. For example, consider the algebraic equation obtained from our original proportion:

• \(15x - 945 = 8x\)

This equation results from cross-multiplying and expanding terms. By simplifying this equation, we rearrange it to find the solution. Understanding how to manipulate algebraic expressions is crucial in solving such problems. In algebra, our goal is to isolate the unknown variable on one side of the equation, allowing us to easily solve for it. Always follow these key steps to keep your equation balanced:

• Combine like terms
• Use inverse operations to isolate the variable
• Simplify your solution

Algebraic equations form the backbone of many mathematical concepts. Mastering these basics will enhance your problem-solving skills across various disciplines.