Problem 33 - Algebra for College Students

Question

Solve each equation for \(x\). \(\frac{x-a}{b}=c\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = bc + a\).

Step 1: Understand the Equation

The given equation is a linear equation in terms of variable \(x\) and constants \(a\), \(b\), and \(c\). The equation is written as \(\frac{x-a}{b} = c\).

Step 2: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the equation by \(b\). This gives: \[(x - a) = b \cdot c\].

Step 3: Simplify the Equation

Expand the right-hand side of the equation by multiplying \(b\) with \(c\), resulting in:\[x - a = bc\].

Step 4: Solve for x

Add \(a\) to both sides of the equation to isolate \(x\):\[x = bc + a\].

Key Concepts

Solving Equations

When tackling equations, our goal is to find the value of the unknown variable, in this case, \( x \). Solving equations is like solving a puzzle where every piece has its place and purpose. Understanding the equation is the first step.
Next, eliminate any fractions or complex terms to simplify the expression step by step. Often you'll use basic operations to isolate the variable.
This might involve adding, subtracting, multiplying, or dividing both sides of the equation to keep them balanced.

• Identify the variable: Determine what you're solving for. Here, it's \( x \).
• Isolate the variable: Use operations to get the variable by itself on one side of the equation.
• Perform operations equally: Whatever you do to one side, do to the other to maintain equality.

Every little step brings you closer to uncovering the unknown, just like reverse engineering a riddle. Taking your time to understand each part of the equation is crucial for solving it correctly.

Equation Manipulation

Equation manipulation involves re-arranging and simplifying equations to make them easier to solve. Think of it as reshaping clay. You mold it in different ways until you reach the form you desire, which is the simplest version of the equation.
For the given equation \( \frac{x-a}{b} = c \), our goal through manipulation is to eliminate the fraction and simplify it.
Here’s how you can achieve this:

• Eliminate the fraction: Multiply both sides by the denominator \( b \) to clear the fraction. So the equation \( \frac{x-a}{b} = c \) becomes \( x-a = bc \).
• Simplify further: In this simplified form, further rearrange to isolate \( x \), leading us to \( x = bc + a \).

Manipulating equations requires you to clearly see each step and strategically use mathematical operations to tidy up and simplify the equation into its most workable form.

Algebraic Expressions

An algebraic expression involves numbers, variables, and arithmetic operations. It's like a recipe consisting of different ingredients. In our equation, \( \frac{x-a}{b} = c \), the expression \( \frac{x-a}{b} \) comprises variable \( x \) and constants \( a \) and \( b \).
Each part plays a significant role in understanding how the equation functions and how to manipulate it.
Knowing how these components interact helps us understand:

• Variables and constants: Variables like \( x \) change, while constants \( a, b, c \) maintain fixed values.
• Operations (addition, subtraction, division): Think of these as tools that help rearrange and solve the equation.
• Forms of expressions: Expressions can be rearranged to highlight different relationships within the equation.

Algebraic expressions are key to breaking down and reconstructing equations to understand the variable relationships and solve for unknown values effectively.