Problem 77 - Introductory and Intermediate Algebra for College Students 4th

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 \(y-a=-b,\) then \(y=a+b\)

Step-by-Step Solution

Answer

The statement is false. The correct statement should be: 'If \(y-a=-b,\) then \(y=a-b\)'.

Step 1: Analyze the given statement

We have been given the statement 'If \(y-a=-b,\) then \(y=a+b\)'. According to this statement, if we isolate 'y' in the equation \(y-a=-b\), we should get \(y=a+b\).

Step 2: Isolate 'y'

We can isolate 'y' in the equation \(y-a=-b\) by adding 'a' to both sides of the equation. This gives us the equation \(y=a-b\).

Step 3:Determine the truth of the statement

The statement 'If \(y-a=-b,\) then \(y=a+b\)' is therefore false because actually, if \(y-a=-b,\) then \(y=a-b\) instead of \(y=a+b\).

Key Concepts

Equation Solving

In introductory algebra, equation solving is a fundamental skill. It involves finding the value of a variable that makes a given equation hold true. Equations are like puzzles: they show a relationship between different values and you need to find the unknown piece. For instance, if you have an equation like \(y-a=-b\), the goal is to determine what \(y\) should be for the equation to be correct. This requires understanding and manipulating algebraic expressions, often using techniques such as addition, subtraction, or balancing both sides. It's important to perform the same operation on both sides of the equation to maintain equality. This principle ensures that the equation remains valid while you work towards solving it.

Isolating Variables

Isolating a variable involves manipulating an equation such that a particular variable stands alone on one side. This is a vital skill for solving equations. In the given exercise, we were tasked with isolating \(y\) in the equation \(y-a=-b\). To achieve this, we added \(a\) to both sides of the equation. This step is crucial because it helps to simplify the equation and reveal the true relationship between the variables involved.

• Additive or subtractive operations are commonly used, depending on the equation structure.
• Be careful to perform the same operation on both sides to keep the equation balanced.

Isolating variables can make it easier to identify errors in algebraic problem-solving, as seen when the initial statement about \(y\) was found to be incorrect.

Mathematical Truth Statements

In mathematics, a truth statement is a proposition or equation that accurately reflects the relationship between quantities or operations as stated. Evaluating truth statements involves verifying if the proposed relationships and operations hold true. For example, the statement "If \(y - a = -b\), then \(y = a + b\)" was initially proposed. To assess this statement’s validity, we needed to isolate \(y\) and check if the transformation was correct. When isolating \(y\), we found that \(y = a - b\), not \(y = a + b\). This discrepancy revealed that the original statement was false, highlighting the importance of checking and verifying each step in mathematical logic. Uncovering false statements is a valuable skill in algebra, as it enables you to rectify and refine your understanding of mathematical principles. Always double-check your results against the given statement to ensure accuracy.