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Chapter 11: Problem 115

For problems \(57-140\), solve each equation. $$ -5-y=-2 $$

Short Answer

Expert verified
The solution to the equation is \( y = -3 \).

Step by step solution

01

Understand the Equation

The given equation is \[ -5 - y = -2 \]This equation consists of constants and a variable \(y\). Our goal is to solve for \(y\).
02

Isolate the Variable

To isolate \(y\), we need to get \(y\) by itself on one side of the equation. First, add \(5\) to both sides of the equation:\[ -5 - y + 5 = -2 + 5 \]This simplifies to:\[ -y = 3 \]
03

Solve for the Variable

Now, we solve for \(y\) by multiplying both sides of the equation by \(-1\) to get \(y\) positive:\[ -1(-y) = -1(3) \]This gives us:\[ y = -3 \]

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Key Concepts

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

Isolation of Variables
When solving linear equations, a fundamental skill is isolating the variable. This means we need to get the variable alone on one side of the equation while everything else is on the other side. In the equation \[ -5 - y = -2 \]the goal is to have \( y \) by itself on one side to figure out its value.

Start by moving any numbers that are not part of the variable term to the opposite side of the equation.
In this case, add 5 to both sides:
  • On the left, \( -5 + 5 \) cancels out, leaving us with \( -y \).
  • On the right, \( -2 + 5 \) simplifies to \( 3 \).
Now the equation is easier to work with:\[ -y = 3 \]By isolating \( y \), you've simplified the equation and are one step closer to finding the solution.
Arithmetic Operations
To successfully isolate the variable, you'll often need to perform arithmetic operations. These basic calculations help rearrange parts of the equation until the variable is isolated. In our example, the following operations were performed:
  • Adding/Subtracting: When you add 5 to both sides, you are using addition to "cancel out" the -5 next to \( y \). It is important to add the same value to both sides to keep the equation balanced.
  • Multiplying/Dividing: Later on, you multiplied both sides by \( -1 \). This operation flips the sign of both sides, allowing you to change a negative \( y \) to a positive \( y \).
These operations keep the equation balanced and ensure you are manipulating the equation correctly, leading to the right solution.
Negative Coefficients
Dealing with negative coefficients may seem tricky, but it becomes simple if you know how to handle them. A coefficient is a number that is multiplied by a variable. In the expression \(-y\), the negative sign can be considered as a coefficient of -1.

To eliminate a negative coefficient, you can multiply or divide each side of the equation by -1. In our example:
  • When we started with \(-y = 3\), \(y\) had a coefficient of -1.
  • Multiplying both sides by \(-1\) flips the signs:\[-1(-y) = -1(3)\], resulting in \(y = -3\).
By understanding how to handle negative coefficients correctly, you can easily solve for the variable, making your calculations simpler and preventing errors.

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