Problem 94 - Introductory Algebra for College Students

Question

Explain how to solve a linear equation containing fractions.

Step-by-Step Solution

Answer

To solve a linear equation with fractions, multiply each term by the largest denominator to remove the fractions and simplify the equation. Once the equation has been simplified, solve the equation in the normal way.

Step 1: Understanding the problem

In order to solve a linear equation containing fractions, first you need to understand what a linear equation is. A linear equation is an equation that makes a straight line when it's graphed. Some examples include \(y = 3x + 2\), \(4y - 2x = 1\), and \(2x = 3 - y\). A fraction involves a numerator (the top number) divided by a denominator (the bottom number). For instance, a linear equation containing fractions could be \(\frac{1}{2}x + \frac{3}{4} = \frac{5}{6}\).

Step 2: Removing Fractions

To solve a linear equation containing fractions, you need to remove the fractions. This can be done by multiplying every term by the denominator of the largest fraction. For example, if our equation is \(\frac{1}{2}x + \frac{3}{4} = \frac{5}{6}\), the term with the largest denominator is the last term on the right side of the equation, \(\frac{5}{6}\). Therefore, we multiply every term in the equation by 6, which leads us to \(3x + 18 = 5\).

Step 3: Solving the Simplified Equation

Now, we are left with a simple linear equation to solve. First perform the subtraction: \(3x = 5 - 18\). Then solve for \(x\) by dividing by 3 from both sides to get \(x = \frac{-13}{3}\).