Question

If \(x+y=0\), which of the following must be equivalent to \(x-y\) ? A) \(-2 y\) B) \(\frac{x}{y}\) C) \(x\) D) \(x^2\)

Short answer

The correct answer is option A) \(-2y\).

Step-by-step solution

Analyze the given equation

To solve the problem, let's first rewrite the given equation (\(x + y = 0\)) into a more useful form. Since we want to find an expression that is equivalent to \(x - y\), we can try to solve the given equation for x and then substitute the result into the expression \(x - y\).

Solve the equation for x

Given the equation \(x + y = 0\), let's solve it for x: \(x = -y\)

Substitute the result in the expression

Now we can substitute the expression we found for x into the expression \(x - y\): \((-y) - y\)

Simplify the resulting expression

After substituting, let's simplify the expression: \(-y - y = -2y\) Our simplified expression is now \(-2y\). Now we can match this simplified expression to one of the answer options.

Compare with the given options

We have found that the expression equivalent to \(x - y\) is \(-2y\). When we look at the given options, this expression matches with option A: A) \(-2y\) So, the equivalent expression for \(x - y\) is \(-2y\), which means the correct answer is option A.

Key concepts

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They often involve variables, which are symbols that stand for unknown values. In our exercise, the equation given is \( x + y = 0 \). This equation means that the sum of \( x \) and \( y \) equals zero. The key idea in solving algebraic equations is to rearrange the terms to express one variable in terms of the other variables or constants in the equation.
For example, when we have \( x + y = 0 \), we can rearrange it to solve for \( x \) by isolating \( x \) on one side of the equation. Here, it becomes \( x = -y \). This step is crucial because it helps us understand the relationship between \( x \) and \( y \), which can then be used in other problems or expressions that involve these variables.

Expression Simplification

Simplifying expressions is an essential step in problem-solving, especially in algebra. It involves rewriting an expression in a simpler and more manageable form. After deriving the relationship \( x = -y \) from the equation \( x + y = 0 \), we need to simplify the expression \( x - y \).

To do this, substitute \(-y\) for \(x\) in the expression:

• \((-y) - y\)

Next, combine like terms by adding these two \(-y\) values. This gives us:

• \(-y - y = -2y\)

By simplifying the expression \(x - y\) as \(-y - y\), we reach a more basic form, \(-2y\). This simplification makes it easier to match against potential solutions in a multiple-choice question. Thus, simplifying expressions can help in quickly identifying equivalent forms and finding the right solution.

Problem Solving Steps

Problem-solving in mathematics often follows a series of logical steps, each aimed at simplifying the problem to arrive at a solution. The SAT problem in question provides a perfect illustration of this process. Let's go over the core steps:

• Understand the Problem: Begin by analyzing the given equation \(x + y = 0\) and identify what needs to be found—in this case, an equivalent expression for \(x - y\).
• Solve the Equation: Rearrange the equation to express one variable in terms of another. We solved for \(x\) to get \(x = -y\).
• Substitute and Simplify: Replace variables accordingly in the expression \(x - y\) and simplify it to find \(-y - y = -2y\).
• Compare and Choose: Match the simplified expression to the given options to select the equivalent form. The equivalent option is A) \(-2y\).

By breaking down the problem into these steps, students can effectively solve similar algebraic problems. Understanding each step ensures that learners approach algebraic expressions methodically, reducing errors and improving their problem-solving skills.