Problem 24 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{x+1}{x-1}=\frac{6}{4} $$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Cross Multiply

To solve the proportion \( \frac{x+1}{x-1} = \frac{6}{4} \), first cross multiply the fractions. This gives us the equation: \((x+1) \times 4 = (x-1) \times 6\).

Step 2: Distribute and Simplify

Expand both sides by distributing multiplication: - Left side: \(4(x+1) = 4x + 4\)- Right side: \(6(x-1) = 6x - 6\).The equation is now \(4x + 4 = 6x - 6\).

Step 3: Combine Like Terms

To solve for \(x\), first collect all terms involving \(x\) on one side and constant terms on the other. Subtract \(4x\) from both sides: \(4x + 4 - 4x = 6x - 4x - 6\), which simplifies to \(4 = 2x - 6\).

Step 4: Isolate the Variable

Add 6 to both sides to isolate terms with \(x\): \(4 + 6 = 2x - 6 + 6\), simplifying to \(10 = 2x\).

Step 5: Solve for \(x\)

Divide both sides by 2 to solve for \(x\): \(\frac{10}{2} = x\), which simplifies to \(x = 5\).

Key Concepts

Cross Multiplication Explained

Cross multiplication is a fundamental technique used to solve proportions, which are equations that state two ratios are equal. Let's say you have an equation like \( \frac{a}{b} = \frac{c}{d} \). Cross multiplication involves multiplying the numerator of one ratio by the denominator of the other. This means you'll multiply \( a \) by \( d \) and \( b \) by \( c \). This gives you the equation:

• \( ad = bc \)

This step is crucial in converting a proportion, which is a fraction equation, into a simple equation without fractions. In our example, we cross multiplied to get:

• \((x+1) \times 4 = (x-1) \times 6\)

This translates the proportion into a more manageable equation that we can work with in the next steps.

Understanding Distribution in Algebra

Distribution is an essential algebraic skill that involves multiplying a single term by two or more terms inside parentheses. Imagine you have an expression like \( a(b + c) \). To distribute, you multiply \( a \) by both \( b \) and \( c \), which gives:

• \( ab + ac \)

In our original problem, after cross-multiplying, we need to distribute to eliminate the parentheses:

• Left side: \( 4(x + 1) = 4x + 4 \)
• Right side: \( 6(x - 1) = 6x - 6 \)

By doing this, we've simplified each side of the equation, making it easier to solve. The distribution property is a powerful tool that helps break down complex expressions into more straightforward terms.

The Process of Combining Like Terms

Combining like terms is a method to simplify equations by merging terms that have the same variable component. Consider an expression like \( 3x + 5x \). Both terms are like terms because they contain the same variable, \( x \). By combining them, you get:

• \( 8x \)

In our equation, after distributing, we have \( 4x + 4 = 6x - 6 \). To combine like terms, we rearrange the equation to bring all \( x \) terms on one side and constants on the other:

• Subtract \( 4x \) from both sides: \( 4x + 4 - 4x = 6x - 4x - 6 \)
• This simplifies to \( 4 = 2x - 6 \)

Combining like terms is crucial for simplifying equations and is a pivotal step in solving for the variable.

Isolating the Variable for Solution

Isolating the variable means getting the variable alone on one side of the equation to find its value. This involves several steps of inverse operations like addition, subtraction, multiplication, or division. In the equation \( 4 = 2x - 6 \), we first isolate \( 2x \) by getting rid of the constant:

• Add 6 to both sides: \( 4 + 6 = 2x - 6 + 6 \)
• This simplifies to \( 10 = 2x \)

The next step is to solve for \( x \) by dividing both sides by 2:

• \( \frac{10}{2} = x \)
• So, \( x = 5 \)

These steps of isolating the variable ensure we calculate the exact value that makes the original equation true. Isolating the variable is a cornerstone of algebra that simplifies and solves equations efficiently.