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Chapter 7: Problem 32

Solve. $$4-y-1=0$$

Short Answer

Expert verified
The solution is \(y = 3\).

Step by step solution

01

Simplify the Equation

Begin by simplifying the left side of the equation. Combine the constants to make the equation simpler. The equation is \(4 - y - 1 = 0\). Combine \(4\) and \(-1\) to get \(3\), so the equation now becomes \(3 - y = 0\).
02

Isolate the Variable

The next step is to solve for the variable \(y\). To do this, you need to isolate \(y\) on one side of the equation. Take \(3\) to the other side by subtracting \(3\) from both sides of the equation: \(3 - y - 3 = 0 - 3\), which simplifies to \(-y = -3\).
03

Solve for the Variable

To solve for \(y\), multiply both sides of the equation by \(-1\) to eliminate the negative sign: \(-1 \times -y = -1 \times -3\), which simplifies to \(y = 3\). This is the solution to the equation.

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Key Concepts

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

Equation Simplification
Equation simplification is all about making an equation easier to work with. In the given exercise, we start with the equation \(4 - y - 1 = 0\). To simplify, focus on combining like terms. In this case, \(4\) and \(-1\) are constants. Combining them gives us \(3 - y = 0\). Simplification helps us see the equation more clearly, without extra numbers cluttering up the process. In general, this involves:
  • Identifying and combining like terms, such as numbers or variables.
  • Removing any unnecessary complexities.
A simpler equation makes it easier to spot the next steps needed to solve it.
Isolation of Variables
Isolation of variables is crucial when solving linear equations. The goal is to get the variable by itself on one side of the equation. Let's look at our simplified equation: \(3 - y = 0\). We want to isolate \(-y\) on one side. To do this, we subtract \(3\) from both sides, leading to the equation \(-y = -3\).

Isolation involves:
  • Performing the same operation on both sides of the equation to keep it balanced.
  • Moving all terms with the variable to one side and constants to the other.
This step is like preparing the stage for the variable, making sure it stands alone so we can easily find its value.
Negative Sign Manipulation
Manipulating negative signs often causes confusion, but it's simpler than it seems. Once you've isolated the variable, you may find your equation like this: \(-y = -3\). We need to deal with the negative sign in front of \(y\).

To remove it, we multiply both sides by \(-1\). This step is often referred to as making the variable positive. So \(-1 \times -y = y\) and \(-1 \times -3 = 3\). This gives us \(y = 3\), which is our solution.

Key points to remember:
  • Multiplying both sides of an equation by \(-1\) changes the sign of both sides.
  • This step is essential in finding the positive form of a variable, especially in cases where it appears negative.
Understanding how to handle negative signs ensures you solve equations correctly every time.

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