Question

Solve the equation by using the LCD. Check your solution(s). $$\frac{3}{2}+\frac{1}{x}=2$$

Short answer

The solution to the equation is \(x=2\).

Step-by-step solution

Find the LCD and clear fractions

First, identify the least common denominator (LCD), which is 2x. Multiply every term by this LCD.

After Multiplication

Multiplying the equation through by 2x, we obtain: \(3x+2=4x\).

Simplify the Equation

Simplifying this equation by subtracting 3x from each side, we get: \(x=2\).

Check

We then substitute x = 2 back into the original equation to ensure it holds true. After substitution, we find that it does: \(3/2 + 1/2 = 2 \), hence our solution is valid.

Key concepts

Least Common Denominator

When working with equations that involve fractions, finding the least common denominator (LCD) is a critical step. This method helps you to combine fractions by establishing a common base, which makes simplification easier. The LCD is the smallest number that each denominator in an equation can divide into. For instance, if you're dealing with fractions that have denominators of 2 and x, as in the example equation \(\frac{3}{2}+\frac{1}{x}=2\), the LCD would be 2x because it's the smallest number into which both 2 and x can divide.

To apply the LCD method, multiply each term in the equation by the LCD to eliminate the fractions. This process, often referred to as 'clearing fractions', transforms a fractional equation into a simpler linear equation, which is easier to manage and solve.

Equation Simplification

The simplification of an equation involves combining like terms and reducing the equation to its most basic form. Once you have cleared the fractions using the LCD, the equation becomes more straightforward. For example, the original fraction-based equation \(\frac{3}{2}+\frac{1}{x}=2\) turned into a linear equation \(3x+2=4x\) after multiplying by the LCD of 2x.

Steps to Simplify

• Combine like terms: Gather all the x terms on one side and the constant terms on the other, aiming to isolate the variable.
• Reduce the coefficients: If there are coefficients in front of the variable, divide the entire equation by the coefficient to solve for the variable.
• Finalize the solution: Once the variable is isolated and the equation is simplified, you will have a clear answer.

Following these steps leads to the equation's simplification, making it possible to understand and find the value of the unknown variable.

Solution Checking

After finding a potential solution to an equation, it is paramount to verify its correctness. This step is essential to ensure that no arithmetic or algebraic errors were made during the problem-solving process. To check your solution, substitute the value you found for the variable back into the original equation and perform the operations to see if the equation holds true.

In our exercise, after simplifying the equation, we found that x equals 2. We then plug x=2 back into the original equation to determine its validity: \(\frac{3}{2}+\frac{1}{2}=2\). Simplifying the left side yields 2, which matches the right side of the equation, thereby confirming that our solution \(x=2\) is correct. Always remember to perform this crucial step, as it can save you from overlooking mistakes and solidifies your understanding of the solving process.