Problem 61 - Precalculus

Question

Solve each equation for \(x\) in terms of the other letters. $$\frac{a-x}{a-b}-2=\frac{c-x}{b-c}$$

Step-by-Step Solution

Answer

\(x = \frac{-a}{c-a}\)

Step 1: Eliminate Fractions

Multiply every term by the denominators \((a-b)(b-c)\) to remove the fractions. We obtain: \((a-x)(b-c) - 2(a-b)(b-c) = (c-x)(a-b)\).

Step 2: Expand Terms

Expand each term from the equation: \( (a-x)(b-c) = ab - ac - xb + xc \) and \( (c-x)(a-b) = ca - cb - xa + xb \). Now expand \(-2(a-b)(b-c) = -2(ab - ac - b^2 + bc) = -2ab + 2ac + 2b^2 - 2bc \).

Step 3: Combine Like Terms

Combine all the terms on both sides: \( ab - ac - xb + xc - 2ab + 2ac + 2b^2 - 2bc = ca - cb - xa + xb \).

Step 4: Isolate Terms Containing x

Rearrange all terms involving \(x\) to one side: \(-xb + xc - xa + xb = ab - ac - 2ab + 2ac + 2b^2 - 2bc - ca + cb\). Simplify to get \(-xa + xc = -a \).

Step 5: Solve for x

Factor out \(x\) from the terms on the left: \(x(-a + c) = -a\). Solve for \(x\) by dividing both sides by \(-a + c\): \[ x = \frac{-a}{c-a} \].

Key Concepts

Fraction Elimination

When faced with an equation that involves fractions, the first step is often to eliminate these fractions for a simpler equation. This is achieved by multiplying every term in the equation by a common denominator. For our equation, \( \frac{a-x}{a-b} - 2 = \frac{c-x}{b-c} \), the common denominator is \((a-b)(b-c)\).
This process helps to clear the fractions, making it easier to work with the equation later on.

• Step: Multiply all terms by \((a-b)(b-c)\).
• Goal: Simplify the equation by removing fractional parts.

This transforms the equation into \((a-x)(b-c) - 2(a-b)(b-c) = (c-x)(a-b)\). By doing this, you are left with a clear path to further manipulate the equation without the complexity of fractions.

Variable Isolation

Variable isolation is the process of rearranging an equation such that the variable of interest is on one side, often indicated by a single expression or term. To isolate \(x\), we need to collect all terms involving \(x\) together. This often involves moving terms across the equals sign and changing their signs as needed.
In the given equation, after expansion and combining like terms, we focus on isolating \(x\): \(-xb + xc - xa + xb = ab - ac - 2ab + 2ac + 2b^2 - 2bc - ca + cb\).

• Move all terms involving \(x\) to one side.
• Keep x consistently positive on one side to avoid confusion.

The goal is to have \(x\) only on one side of the equation so you can easily solve for it in the next steps.

Expression Expansion

Expanding expressions involves distributing terms within parentheses to eliminate them. It’s essential for simplifying polynomials and other expressions. When you multiply terms like \((a-x)(b-c)\), you expand by distributing each element in the first parenthesis across the second.
For example:

• \((a-x)(b-c)\) becomes \(ab - ac - xb + xc\).
• Do similarly for other products: \((c-x)(a-b)\) and \(-2(a-b)(b-c)\).

This expansion helps in visualizing each individual term, making it easier to combine and eliminate terms later. It's often a necessary precursor to simplifying and finding solutions to algebraic equations.

Combining Like Terms

Combining like terms is a technique used for simplifying expressions. It involves adding or subtracting terms that have the same variable components. Once we've expanded our expressions, the next step is to look for like terms across both sides of the equation and combine them.
In our example, terms like \(-xb\), \(xb\) and others can be combined:

• Combine: \(-xb + xb\)
• Also consider constants and comparable terms on opposite sides.

This step reduces the complexity by minimizing the number of terms, paving the way for variable isolation. A simpler equation is easier to solve, aiding in reaching the solution quicker and with less chance of error.