Problem 80 - 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. $$\text { If } A=l w, \text { then } w=\frac{l}{A}$$

Step-by-Step Solution

Answer

The given statement 'If \(3x=18\), then \(x=18-3\)' is false. The corrected true statement would be 'If \(3x=18\), then \(x=18/3\)'.

Step 1: Input the given statement

Consider the given statement - If \(3x=18\), then \(x=18-3\). Our task is to confirm if it's true or false.

Step 2: Analyze the initial equation

The initial part of the equation is \(3x=18\). In this case, we are to isolate \(x\). To do this, we divide both sides of the equation by 3, which gives us \(x=18/3 = 6\).

Step 3: Verify the latter part of the statement

The latter part of the statement is \(x=18-3\). This simplifies to \(x=15\). Comparing this to the value from our initial equation (\(x=6\)), we can see there is a contradiction.

Step 4: Correcting the false statement

Since the given statement is false, it needs to be corrected. The correct statement would be - If \(3x=18\), then \(x=18/3\).

Key Concepts

Equation Solving

Equation solving is a fundamental aspect of algebra and mathematics as a whole. It involves finding the value of the variable that makes the equation true. In other words, we seek the number that, when substituted for the variable, satisfies the equation.

When solving equations, particularly simple linear equations like the one in our exercise, one common strategy is to isolate the variable on one side of the equation. This can be done through various operations such as:

• **Addition or subtraction** - To eliminate constants on one side of the equation.
• **Multiplication or division** - To remove coefficients from the variable.

For instance, in the problem where we have the equation \(3x = 18\), the goal is to have \(x\) by itself on one side. Here, dividing both sides by 3 simplifies to \(x = 6\). This solution accurately represents the value of \(x\) that satisfies the equation.

Literal Equations

Literal equations are equations that consist of several variables instead of only numerical values. These equations are commonly used in various applications as they allow one to solve for one variable in terms of other variables. They are especially useful in formulas used in science and engineering.

While the exercise focuses on an equation with only one variable, the process of solving for a variable includes techniques that are pivotal in handling literal equations too. Let's say you have an equation like \(Ax = B\). To solve for \(x\), you would perform division, resulting in \(x = \frac{B}{A}\). This reveals the general approach to dealing with literal equations: manipulate one side of the equation to have only the desired variable. Understanding this principle can significantly enhance your ability to tackle more complex problems involving multiple variables.

Mathematical Statements

Mathematical statements are declarations that can be either true or false. They are the building blocks for mathematical reasoning and problem-solving.

In mathematics, verifying whether a statement is true or false is essential before arriving at a solution. For instance, our original exercise involves determining if the statement "If \(3x = 18\), then \(x = 18 - 3\)" is true. Through verification, we established that this statement is false. The correct transformation needed for equation solving is replacing subtraction with division, leading to the true statement \(x = \frac{18}{3}\).

Understanding how to evaluate and adjust mathematical statements effectively is crucial. It allows learners to engage deeply with the content and develop a robust comprehension of logical processes within mathematics. Through mastering these skills, students are better prepared to identify inaccuracies and correct them accurately.