Problem 38 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-2 y-8=2$$

Step-by-Step Solution

Answer

The solution for \(y\) is \(-5\).

Step 1: Understand the Equation

We are given the equation \(-2y - 8 = 2\). Our goal is to isolate \(y\) by using algebraic operations while maintaining the equality.

Step 2: Use the Addition Property of Equality

To eliminate the constant term on the left side of the equation, add 8 to both sides. This will help in isolating the term containing \(y\). \(-2y - 8 + 8 = 2 + 8\) Simplifying gives: \(-2y = 10\).

Step 3: Isolate the Variable y

Now, divide both sides of the equation by \(-2\) to solve for \(y\).\(\frac{-2y}{-2} = \frac{10}{-2}\) Simplifying gives: \(y = -5\).

Key Concepts

Solving Equations

When we talk about solving equations in algebra, we refer to finding the value of an unknown variable that makes the equation true. It’s like unlocking a puzzle where the key is to make both sides of the equation equal.
The fundamental principle behind solving equations is to perform the same mathematical operation on both sides. This ensures that the equality holds, and the balance between the two sides is maintained. In this particular equation, \(-2y - 8 = 2\), solving it means finding the value of \(y\) that balances the equation.
Overall, solving equations is a structured series of steps. These steps often involve eliminating numbers and variables using algebraic operations such as addition, subtraction, multiplication, or division.

Isolating Variables

Isolating variables is crucial when solving equations. It involves manipulating the equation to get the variable, in this case \(y\), all by itself on one side of the equation. This process helps us figure out the value of the variable.
To isolate \(y\) in \(-2y - 8 = 2\), we started by eliminating the constant \(-8\) from the left side, using the addition property of equality. This property says that you can add the same number to both sides of the equation without changing its truthfulness.

• Adding 8 to each side gave us: \(-2y - 8 + 8 = 2 + 8\), simplifying to \(-2y = 10\).
• Next, dividing by \(-2\) isolates \(y\).

These steps are essential as they systematically isolate \(y\), uncovering its value.

Algebraic Operations

Algebraic operations are the tools used to transform and solve equations. They consist of operations like addition, subtraction, multiplication, and division. When applied correctly, they maintain the balance of the equation while bringing us closer to finding the unknown variable.
In the example \(-2y - 8 = 2\), we first used the addition operation to tackle the subtraction of 8. This operation nullified the effect of \(-8\) on \(-2y\), transforming it into a simpler equation, \(-2y = 10\).
Following this, division was used to solve for \(y\). By dividing each side by \(-2\), the term \(-2y\) became \(y\), leading us directly to \(y = -5\).
These algebraic operations, when used systematically, make solving equations efficient and straightforward.