Problem 73 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(a\) is any real number, the equation \(\frac{a}{x}+1=\frac{a}{x}\) has no solution.

Step-by-Step Solution

Answer

True. The statement is correct that 'If \(a\) is any real number, the equation \(\frac{a}{x}+1=\frac{a}{x}\) has no solution'

Step 1: Simplify the equation

The equation given in the statement is \(\frac{a}{x}+1=\frac{a}{x}\). Let's start simplifying this. Subtract \(\frac{a}{x}\) from both sides to isolate the '1'. So, the equation simplifies to 1=0.

Step 2: Check if equation is possible

Looking at the equation, 1=0, it is clear that this equation is impossible. The value 1 cannot equal 0.

Step 3: Determine the truthfulness of the statement

Since the simplified equation is impossible, it means that there are no values of \(x\) and \(a\) that can satisfy the original equation (\(\frac{a}{x}+1=\frac{a}{x}\)). Thus, the statement saying 'if \(a\) is any real number, the equation \(\frac{a}{x}+1=\frac{a}{x}\) has no solution' is true.

Key Concepts

Understanding Equations

Equations are mathematical statements that express the equality between two expressions. They play a vital role in algebra by allowing us to solve for unknown variables. In our original exercise, the equation is \(\frac{a}{x} + 1 = \frac{a}{x}\). Here, you have two expressions set equal to each other, which signifies a balance that should be analyzed. Solving an equation usually involves finding a value for the variable that makes this statement true. In our case, we need to find out if there is a value for \(x\) or \(a\) that can satisfy the equation. However, by rearranging and simplifying, this equation leads to an untrue equation where 1 equals 0, showing that it has no solution.

The Art of Simplification

Simplification in algebra involves making an equation or expression as straightforward as possible. This can involve techniques such as combining like terms, factoring, or eliminating unnecessary parts of an equation. In this exercise, we start by simplifying \(\frac{a}{x} + 1 = \frac{a}{x}\) by subtracting \(\frac{a}{x}\) from both sides.

By doing this, the equation reduces to 1 = 0, which is a contradiction. Simplification helps in identifying whether an equation is solvable by revealing its core components clearly. When simplification leads to a statement that is obviously untrue, as we have here, it indicates that the original equation has no solution. This highlights how critical simplification is in solving algebraic problems.

Understanding Real Numbers

Real numbers are all the numbers that can be found on the number line, including both rational numbers (like fractions) and irrational numbers (like the square root of 2 or \(\pi\)). They are fundamental to most areas of mathematics, including algebra. Real numbers are crucial when solving equations, as they provide the potential values that variables like \(x\) and \(a\) might take.

In this exercise, it is stated that if \(a\) is any real number, the original equation \(\frac{a}{x} + 1 = \frac{a}{x}\) has no solution. This reflects the fact that no matter what real number \(a\) represents, subtracting \(\frac{a}{x}\) from both sides resulted in the unsolvable equation 1 = 0. Understanding the properties of real numbers helps to comprehend why certain equations might result in no solutions, as no tangible value from the set of all real numbers will satisfy the condition.