Problem 18 - College Algebra Essentials

Question

Contain linear equations with constants in denominators. Solve equation. \(\frac{x}{5}=\frac{x}{6}+1\)

Step-by-Step Solution

Answer

The solution to the equation is x = 30.

Step 1: Eliminate Denominators

Multiply both sides of the equation by 30 (which is the least common multiple of 5 and 6) to eliminate the denominators. This gives us \(6x = 5x + 30\).

Step 2: Simplify Equation

Subtract 5x from both sides of the equation to get \(x = 30\).

Step 3: Check Solution

Substitute x = 30 back into the original equation to check if the left hand side equals the right hand side. \( \frac{30}{5} = \frac{30}{6} + 1 \) simplifies to \(6 = 5 + 1\), which verifies the solution.

Key Concepts

Solving Equations

Solving equations is all about finding the value of the unknown that makes the equation true. In this exercise, the unknown is represented by the variable \( x \). To solve an equation, you need to perform operations that simplify the expression, gradually isolating \( x \) on one side of the equation.
Here’s a simple plan you can follow:

• Look at both sides of the equation and identify terms that can be simplified or moved around.
• Use operations like addition, subtraction, multiplication, and division to rearrange and simplify terms, step by step.
• As you perform these operations, remember to do the same thing to both sides of the equation to maintain balance.

In our exercise, after eliminating the denominators, we subtracted \( 5x \) from each side to find \( x \).

Least Common Multiple

Finding the least common multiple (LCM) is key when working to eliminate denominators in equations like fractional linear equations. The LCM of a set of numbers is the smallest number that each of the numbers divides into evenly.
Why find the LCM?

• It helps to remove fractions by giving us a common denominator, making it easier to work with whole numbers.
• It simplifies the arithmetic process and reduces the chance for errors.

In our exercise, the LCM of 5 and 6 is 30, enabling us to multiply through and work with a simpler equation. When you multiply each term by 30, the denominators vanish, transforming the fractional terms into integers. This makes solving for \( x \) straightforward.

Eliminating Denominators

Eliminating denominators is a useful technique in solving equations that contain fractions. It simplifies complex equations and allows you to work with whole numbers instead of fractions.
Here's how to eliminate denominators:

• Determine the Least Common Multiple (LCM) of all denominators in the equation.
• Multiply each term in the equation by this LCM. This action will cancel out the denominators.
• Once denominators are gone, you can solve the equation using simple arithmetic operations.

In the given equation \( \frac{x}{5} = \frac{x}{6} + 1 \), multiplying each side by 30 removed the denominators \( 5 \) and \( 6 \), simplifying it to \( 6x = 5x + 30 \). This process makes finding the solution easier!