Problem 83 - Algebra and Trigonometry

Question

\(79-92\) Solve the equation for the indicated variable. $$ \frac{a x+b}{c x+d}=2 ; \quad \text { for } x $$

Step-by-Step Solution

Answer

The solution for \( x \) is \( x = \frac{2d - b}{a - 2c} \).

Step 1: Cross-multiply

Start by cross-multiplying to eliminate the fraction from the equation. Multiply both sides by \(cx + d\):\[ a x + b = 2(c x + d) \].

Step 2: Distribute and simplify

Distribute the 2 on the right side of the equation:\[ a x + b = 2cx + 2d \].

Step 3: Arrange terms for x on one side

Move terms involving \( x \) to one side and constant terms to the other side:\[ a x - 2cx = 2d - b \].

Step 4: Factor out x

Factor \( x \) out of the left side of the equation:\[ x(a - 2c) = 2d - b \].

Step 5: Solve for x

Divide both sides by \( a - 2c \) to isolate \( x \):\[ x = \frac{2d - b}{a - 2c} \].

Key Concepts

Solving for a Variable

When solving algebraic equations, finding the value of a variable is often the goal. Here, the task is to identify the value of \( x \) that satisfies the equation. To do this efficiently:

• Understand what the problem is asking: Find \( x \).
• Follow the basic steps used in algebra: simplify, manipulate, and solve for \( x \).
• Ensure all terms involving the variable are on one side, which makes it easier to solve.

These strategies are consistently applicable, regardless of how complex the equation may initially appear. By breaking down each step, you can systematically solve for the desired variable.

Cross-Multiplying

Cross-multiplying is a technique used to eliminate fractions. It makes equations easier to work with by simplifying the mathematical expressions. In equations like \(\frac{ax+b}{cx+d} = 2\), cross-multiplication involves multiplying the numerator of one side by the denominator of the other, and vice versa. Here's how it works:

• First, identify the fractions in the equation.
• Multiply both sides of the equation by \( cx + d \) to remove the fraction.
• Replace the equation with the simplified version: \( ax + b = 2(cx + d) \).

This transforms a fractional equation into a more manageable format, laying the groundwork for further simplification.

Factoring Expressions

Factoring is a crucial skill in algebra, allowing you to simplify expressions and solve equations more easily. In the context of our exercise, once we have rearranged terms such as \( ax - 2cx = 2d - b \), we can use factoring to simplify further. Here's the process:

• Observe the terms involving \( x \) on one side: \( ax - 2cx \).
• Notice that \( x \) is common in both terms, allowing it to be factored out.
• Rewrite the expression as \( x(a - 2c) \).

Factoring not only simplifies the equation but is also a critical step before isolating the variable. It reduces complex expressions to simpler forms, bringing you closer to the solution.

Isolating Variables

Isolating a variable is often the final step in solving an equation. It involves getting the variable on one side of the equation, leaving it by itself to determine its value. In this case, after factoring, we have \( x(a - 2c) = 2d - b \). To isolate \( x \):

• Recognize that \( x \) is multiplied by \( a - 2c \).
• To isolate \( x \), divide every term on both sides of the equation by \( a - 2c \).
• This gives the solution: \( x = \frac{2d - b}{a - 2c} \).

Isolating the variable is a straightforward process once all other steps are completed. It helps in determining the exact value of the variable you are solving for.