Problem 82 - 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. Solving \(x-y=-7\) for \(y\) gives \(y=x+7\)

Step-by-Step Solution

Answer

The statement 'Solving \(x-y=-7\) for \(y\) gives \(y=x+7\)' is true.

Step 1: Analyze the Original Equation

The original equation given is \(x-y=-7\). We are tasked with solving this equation for \(y\), something that involves isolating \(y\) on one side of the equation.

Step 2: Isolate y

To isolate \(y\), we need to eliminate \(x\) from the left side of the equation. This can be done by adding \(y\) to both sides and then subtracting 7 from both sides of the equation. This gives us \(y=x+7\).

Step 3: Compare the Resulting Equation

We compare the calculated equation \(y=x+7\) with the statement we were given to verify. They match exactly, meaning the original statement is true.

Key Concepts

Linear Equations

Linear equations are mathematical expressions representing a straight line when graphed. They usually take the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These equations are called linear because they describe a linear relationship between two variables, which remains constant throughout. When you have a linear equation, the main goal is to find the value of the variable that makes the equation true. This involves rearranging or simplifying the equation to isolate the variable on one side.

• Linear equations can represent real-world scenarios, like calculating expenses or determining trends.
• Understanding the properties of linear equations can help in various fields like physics and engineering.

Being familiar with forming and solving these simple equations lays a foundation for more complex mathematical concepts.

Solving Equations

Solving equations involves finding the value of the variable that satisfies the equation. When we talk about solving, it means applying certain mathematical operations to both sides of the equation to maintain balance while isolating the variable. Consider the equation \(3x + 7 = 0\). To solve for \(x\), we need to perform steps that paint a clear picture of how each change affects the equation. Apply the inverse of the operations present:

• Subtract 7 from both sides: \(3x + 7 - 7 = 0 - 7\), simplifying to \(3x = -7\).
• Divide both sides by 3: \(x = -\frac{7}{3}\).

The key is maintaining an equivalent balance in the equation throughout the process. Each operation should help edge closer to isolating the variable.Insights from solving linear equations include understanding the value of balancing actions like addition, subtraction, multiplication, and division applied equally to both sides.

Mathematical Reasoning

Mathematical reasoning is the logical thought process which is crucial when dealing with equations. It's about understanding not just the 'how' but the 'why' behind every step taken in solving an equation. When evaluating whether a statement like \("x = \frac{7}{3}\"\) is true, reasoning helps us question and check our solutions. Through reasoning, we:

• Verify each step to ensure logical consistency.
• Avoid common pitfalls like sign errors or miscalculations, crucial in deciding if a solution holds true.

In our exercise, reasoning indicated that the claim was false because solving the equation \(3x + 7 = 0\) actually leads to \(x = -\frac{7}{3}\). This kind of reasoning skill is pivotal not just in mathematics, but also in decision making in everyday life, contributing to critical thinking skills and the ability to analyze problems deeply.