Problem 30 - Elementary Algebra

Question

\(\frac{4}{x-2}=\frac{5}{x+6}\)

Step-by-Step Solution

Answer

x = 34

Step 1: Cross-Multiply

To solve the equation \( \frac{4}{x-2} = \frac{5}{x+6} \), we first cross-multiply: \( 4(x+6) = 5(x-2) \). This transforms the equation into a format that's easier to solve by eliminating the fractions.

Step 2: Distribute Terms

Next, distribute the constants across the terms in the parentheses: \( 4 \times x + 4 \times 6 = 5 \times x - 5 \times 2 \), which simplifies to \( 4x + 24 = 5x - 10 \).

Step 3: Solve for x

Now, move all terms containing \( x \) to one side of the equation and constant terms to the other side: \( 4x + 24 - 5x = -10 \). Simplify this to \( -x + 24 = -10 \). Then, add \( x \) to both sides to get \( 24 = x - 10 \).

Step 4: Isolate x

To isolate \( x \), add 10 to both sides of the equation: \( 24 + 10 = x \), resulting in \( x = 34 \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a valuable technique for solving rational equations, often simplifying the process and making it easier to work with. In our given exercise, \( \frac{4}{x-2} = \frac{5}{x+6} \), cross-multiplication was used to eliminate fractions. This method involves multiplying the numerator of each fraction by the denominator of the other.Here's a step-by-step breakdown:

• Start with the equation: \( \frac{4}{x-2} = \frac{5}{x+6} \).
• Cross-multiply by multiplying \( 4 \) (the numerator of the left fraction) by \( (x+6) \) (the denominator of the right fraction).
• Similarly, multiply \( 5 \) (the numerator of the right fraction) by \( (x-2) \) (the denominator of the left fraction).
• You end up with the equation: \( 4(x+6) = 5(x-2) \).

By transforming the equation this way, you eliminate the fractions and create a linear equation. This allows for easier manipulation of terms and sets the stage for solving the equation directly.

Isolation of Variables

Isolation of variables is a foundational concept in solving equations. The goal is to "isolate" the variable (in this case, \( x \)) on one side of the equation to determine its value. Starting from the already simplified equation after using cross-multiplication and distribution, \( 4x + 24 = 5x - 10 \), the focus shifts to moving terms around.To isolate \( x \):

• Move variable terms together by subtracting \( 5x \) from both sides, resulting in \( 4x - 5x + 24 = -10 \).
• This simplifies to \( -x + 24 = -10 \).
• Next, deal with the constant term by subtracting 24 from both sides, which gives \( -x = -34 \).
• Finally, multiply both sides by \(-1\) to solve for \( x \): \( x = 34 \).

The isolation process clarifies the final value of \( x \), neatly wrapping up the solution. This step is crucial when working with equations, ensuring you're left with a clear and direct expression for your variable.

Distributive Property

The distributive property simplifies expressions by spreading multiplication over addition or subtraction within parentheses. This property is crucial for handling terms effectively, as seen in the solution of our given problem after cross-multiplication.In this scenario, the distributed property is applied as follows:

• Start with the equation after cross-multiplying: \( 4(x+6) = 5(x-2) \).
• Distribute \( 4 \) across \( (x+6) \): \( 4 \times x + 4 \times 6 \) results in \( 4x + 24 \).
• Similarly, distribute \( 5 \) across \( (x-2) \), creating \( 5 \times x - 5 \times 2 \), which simplifies to \( 5x - 10 \).

Using the distributive property transforms complicated expressions into manageable terms, enabling the simplification needed to isolate and solve for \( x \). Understanding this property is essential, as it is widely used to break down and solve mathematical problems efficiently.