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Chapter 3: Problem 61

Solve each equation for y. See Section 2.5. $$ x+y=5 $$

Short Answer

Expert verified
The solution is \( y = 5 - x \).

Step by step solution

01

Identify the Objective

We want to solve the equation for the variable \( y \). This means we need to isolate \( y \) on one side of the equation.
02

Rearrange the Equation

Start with the given equation \( x + y = 5 \). To isolate \( y \), we need to move \( x \) to the other side of the equation.
03

Subtract x from Both Sides

Subtract \( x \) from both sides of the equation to get \( y \). This gives us: \( y = 5 - x \).By subtracting \( x \), we have successfully isolated \( y \) on the left side.

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Key Concepts

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

Isolating Variables
Isolating a variable in an equation means solving for that variable. We aim to get the variable by itself on one side of the equation, separating it from other terms. In our example, we wanted to solve for \( y \) in the equation \( x + y = 5 \). To do this, we needed \( y \) alone on one side of the equation.

Here's a simplified approach to isolating a variable:
  • Identify what variable you need to isolate.
  • Decide which operations will help you separate the variable from other terms.
  • Apply these operations to both sides of the equation to maintain equality.
By following these steps, you maintain the balance of the equation while moving unwanted terms away from the variable you are focusing on.
Equation Rearrangement
Equation rearrangement involves changing the position of terms to isolate the variable of interest. In the given problem, we started with the equation \( x + y = 5 \). Our goal was to isolate \( y \).

The key steps were:
  • Identifying which term needs to be moved (i.e., \( x \) in this case).
  • Determining how to move it (via subtraction).
By subtracting \( x \) from both sides, we effectively repositioned the terms, ultimately transforming our equation into \( y = 5 - x \). This new rearrangement now presents \( y \) isolated and the equation solved under that context.
Basic Algebra Concepts
Understanding basic algebra concepts is crucial for solving equations like \( x + y = 5 \). Algebra involves manipulating symbols and numbers to uncover unknown values. One foundational concept is maintaining equation balance.

When you perform an operation on one side of an equation, like subtracting \( x \), you must do the same to the other side to keep it balanced. This concept ensures that the equation remains valid and true.
  • Maintaining balance is key: Every operation on one side must be balanced by the same operation on the other side.
  • Variables represent unknown values, which you find through isolation and rearrangement.
Mastering these concepts allows you to methodically break down and solve problems using critical thinking techniques.

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