Problem 72 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$\frac{x}{4}+3=\frac{x}{4}$$

Step-by-Step Solution

Answer

The equation has no solutions.

Step 1: Identify Like Terms

Begin this problem by setting the equation in a way that the terms involving \(x\) are on one side of the equation, and the constant terms (if any) are on the other side. The given equation is \(\frac{x}{4}+3 = \frac{x}{4}\).

Step 2: Cancel Like Terms

As \(\frac{x}{4}\) is the same on both sides of the equation, one can subtract \(\frac{x}{4}\) from both sides to cancel it out. This results in \(3 = 0\).

Step 3: Analyze the Result

The resulting statement, \(3 = 0\), is not a valid mathematical statement and therefore, this equation has no solution.

Key Concepts

Algebraic Expressions

In algebra, an algebraic expression is a combination of numbers, variables, and operations like addition and multiplication. For instance, in the equation \(\frac{x}{4}+3=\frac{x}{4}\), \(\frac{x}{4}+3\) and \(\frac{x}{4}\) are algebraic expressions.

Algebraic expressions can range from simple, with just one variable and no operations, to very complex, with multiple variables and various operations. For students to be successful in solving equations, they must first understand how to manipulate these expressions. This involves operations such as combining like terms, using the distributive property, and canceling terms out.

To master this area of algebra, practice simplifying expressions and remember that to maintain equality, what you do to one side of an equation, you must do to the other.

No Solution Equations

Sometimes, an equation may lead to a statement that is not true for any value of the variable involved; these are known as no solution equations. In our initial equation, when we simplified the algebraic expressions and canceled out the like terms, we ended up with the equation \(3=0\), which is never true for any real number.

This type of result indicates that the original equation has no solution. These are important to recognize because they tell us that no real number, when plugged into the variable's place, will satisfy the equation. It's like trying to find a number that makes a false statement true–it's not possible. Recognizing such outcomes is essential as it prevents unnecessary further calculations and allows the student to accurately convey the characteristics of the equation.

Equation Solving Techniques

When it comes to equation solving techniques, there are several strategies that can be employed. The step-by-step solution for the given problem showcases a common technique: identifying and cancelling like terms.

Other techniques include:

• Isolating the variable on one side of the equation.
• Using the distributive property to remove parentheses.
• Combining like terms to simplify the equation.
• Using the properties of equality to perform the same operation on both sides of the equation.

For equations that result in no solution, as in our example, recognizing that the equation simplifies to an impossible statement is crucial. Understanding a variety of solving techniques is advantageous because it prepares students for a wide range of algebraic problems. Regular practice of these techniques, applied in different contexts, reinforces algebraic principles and skill in problem-solving.