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Chapter 2: Problem 34

Solve each equation for \(x\). \(\frac{x}{a}-1=b\)

Short Answer

Expert verified
\( x = ab + a \)

Step by step solution

01

Isolate the Fraction Term

First, we want to isolate the term that contains the variable \(x\). The equation is \( \frac{x}{a} -1 = b \). Add 1 to both sides to get \( \frac{x}{a} = b + 1 \).
02

Eliminate the Denominator

To remove the fraction, multiply both sides of the equation by \(a\). This gives \( x = a(b + 1) \).
03

Simplify

Distribute \(a\) through the parenthesis on the right side of the equation to simplify: \( x = ab + a \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Variables
Variables are like placeholders in mathematics. They often represent unknown numbers that we are trying to find. In the equation \(\frac{x}{a} - 1 = b\), the variable we are dealing with is \(x\). This means we are trying to find the value of \(x\) that makes the equation true.
A variable can be any letter (commonly \(x\), \(y\), or \(z\)) and is used in equations where the actual numbers are not known right away. In this case, \(a\) and \(b\) are constants or known values, while \(x\) is what we need to figure out.
If you are ever in doubt about what a variable represents, think of it as a question mark that you're trying to replace with a number.
Navigating Fraction Equations
When working with fraction equations, our goal is often to eliminate the fraction component. This makes it easier to solve for the variable.
In the equation \(\frac{x}{a} - 1 = b\), the term \(\frac{x}{a}\) is a fraction. This means the unknown variable \(x\) is divided by \(a\). To handle this properly, you will need to isolate this fraction.
  • First, balance the equation by removing any terms that don't contain the fraction. In our case, adding 1 to both sides achieves this.
  • Next, you'll want to cancel out the denominator. You can do this by multiplying both sides by the same number \(a\), which clears the fraction.
This process helps in transforming a complex-looking equation into a simpler one without fractions, making it much easier to solve.
Mastering the Isolation of Terms
Isolating terms is an essential step in solving any equation. It involves rearranging the equation so that the variable you're solving for is alone on one side.
The goal is to tidy up the equation to get something simple like \(x = ...\). This is done through several methods:
  • Adding or subtracting terms from both sides to remove constant numbers.
  • Using multiplication or division to get rid of coefficients or denominators.
In the example equation \(\frac{x}{a} - 1 = b\), the first step in isolating \(x\) is to add 1 to each side, giving \(\frac{x}{a} = b + 1\).
This moves us closer to having \(x\) by itself. By multiplying both sides by \(a\), we eliminate the fraction and find \(x = a(b + 1)\). This process shows how isolating terms simplifies an equation, making it much clearer and easier to solve.

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Most popular questions from this chapter

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