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Chapter 12: Problem 37

Solve each equation for \(y\). $$ x=y+2 $$

Short Answer

Expert verified
The solution is \( y = x - 2 \).

Step by step solution

01

Isolate the Variable

To solve the equation for \( y \), we need to isolate it on one side of the equation. The equation given is \( x = y + 2 \). To isolate \( y \), subtract 2 from both sides of the equation so that \( y \) is by itself on one side. This gives us \( x - 2 = y \).
02

Rearrange the Equation

Once \( y \) is isolated, rewrite the equation for clarity. Instead of having \( x - 2 = y \), position \( y \) on the left side to make it easier to read: \( y = x - 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving equations is like unlocking a mystery. The main goal is to find the value of the unknown variable that makes the equation true. In the exercise, we aim to solve for \( y \) in the equation \( x = y + 2 \). To do this, we follow logical steps that simplify the equation until the mystery variable stands alone. This means adjusting or transforming the equation while always keeping it balanced.
  • An equation is like a balance scale; whatever you do to one side, do to the other.
  • To solve for a variable, carefully apply inverse operations (e.g., if a number is added to the variable, you subtract it to undo the operation).
Once the solution is found, you should double-check by substituting the value back into the original equation. This ensures that the computation is correct.
Isolation of Variables
Isolating a variable is a crucial step in solving equations. This process involves getting the variable we are solving for by itself on one side of the equals sign. In our exercise, isolating \( y \) means making sure that \( y \) stands alone without extra terms attached to it.
  • Start by identifying the variable to isolate. Here, it's \( y \).
  • Remove any numbers or terms on the same side as the variable by doing the opposite operation. Since \( y + 2 \) means 2 is added to \( y \), subtract 2 to isolate \( y \).
In our example, subtracting 2 from both sides of \( x = y + 2 \) results in \( x - 2 = y \). Isolation helps to clarify the relationship between the variables involved.
Equation Rearrangement
Rearranging an equation makes it easier to understand or work with. Following isolation, rearranging is sometimes necessary to present the solution cleanly. It also makes it easier to interpret the relationship between the variables involved.
In our solved equation, \( x - 2 = y \), it might be clearer and more conventional to write it as \( y = x - 2 \). Rearranging equations involves changes like swapping sides or rearranging terms, as long as the equality holds true.
  • It's a good practice to write the variable you're solving for on the left, as this is the standard format.
  • Rearranging helps in checking the solution and makes it transparent for anyone reviewing the work.
While rearrangement doesn't change the solution's correctness, it aligns with mathematical conventions, enhancing clarity and communication of mathematical ideas.

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