Problem 27 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$x-\frac{3}{4}=\frac{9}{2}$$

Step-by-Step Solution

Answer

The solution for the equation \(x-\frac{3}{4}=\frac{9}{2}\) is \(x=\frac{21}{4}\) or \(x=5\frac{1}{4}\)

Step 1: Addition property of equality

First, the goal is to isolate the variable \(x\) on one side of the equation. To achieve that, add \(\frac{3}{4}\) to both sides of the equation. In result, the equation would be: \(x-\frac{3}{4}+\frac{3}{4}=\frac{9}{2}+\frac{3}{4}\)

Step 2: Evaluate both sides

Now, calculate the right side of the equation. To add fractions, each fraction should have the same denominator. Upon further inspection, the denominators of \(2\) and \(4\) are clearly not the same. Convert \(\frac{9}{2}\) into a form that has a denominator of \(4\) by multiplying both the numerator and the denominator by \(2\). Now the equation looks like: \(x=\frac{18}{4}+\frac{3}{4}\). Calculate the right side of the equation: \(x=\frac{21}{4}\)

Step 3: Simplify and Check

After simplifying, the solution for the equation is \(x=\frac{21}{4}\) or as a mixed number, \(x=5\frac{1}{4}\). This solution can be checked by substituting \(x\) with \(\frac{21}{4}\) in the initial equation: if \(\frac{21}{4}-\frac{3}{4}\) equals to \(\frac{9}{2}\), then the solution is correct. After performing the calculation, the left side is indeed equal to \(\frac{9}{2}\), which confirms the solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the unknown variable. In an equation, the expression on one side is equal to the expression on the other side. Solving the equation means finding the value of the variable that makes this statement true.

For instance, let's consider the equation given:

• \(x - \frac{3}{4} = \frac{9}{2}\)

Here, the goal is to isolate the variable \(x\) on one side. Adding \(\frac{3}{4}\) to both sides of the equation is a method used here. This is called the Addition Property of Equality and maintains the equality of the equation while shifting terms from one side to the other.

This technique not only helps in solving simple equations but sets the groundwork for more complex ones involving multiple steps.

Fractions and Decimals

Fractions and decimals often appear in equations, requiring specific techniques for manipulation and simplification. When you encounter fractions in equations, as we do in this exercise, the process involves ensuring each fraction has a common denominator.

Looking at the step given:

• \(x = \frac{18}{4} + \frac{3}{4}\)

By converting fractions to a common denominator, it becomes easier to accurately add or subtract them. Such transformations are necessary because they allow the equation to operate in simple whole fractions, minimizing errors.

Understanding how to convert between fractions and decimals and simplifying the results is essential in keeping equations clear and easy to solve.

Equation Simplification

Equation simplification is often the final step in solving equations. It involves reducing or combining terms to present the solution in its simplest form.

Take the equation obtained after adding fractions:

• \(x = \frac{21}{4}\)

This can be further expressed as a mixed number \(5\frac{1}{4}\), making it easier to interpret. Checking the simplified equation involves substituting the obtained value back into the original equation. Ensuring that both sides equal verifies the solution's accuracy.

In general, equation simplification makes results more understandable, especially when dealing with complex numbers or variables.