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Chapter 2: Problem 22

Solve each equation using the Subtraction and Addition Properties of Equality. $$z-101=-314$$

Short Answer

Expert verified
z = -213

Step by step solution

01

Identify the equation

The given equation is \[ z - 101 = -314 \].
02

Add 101 to both sides

To isolate the variable \( z \), add 101 to both sides of the equation. \[ (z - 101) + 101 = -314 + 101 \]
03

Simplify both sides

Simplify the left side and the right side of the equation: \[ z = -213 \].

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Key Concepts

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

Subtraction Property of Equality
The Subtraction Property of Equality allows you to maintain the balance of an equation when you subtract the same quantity from both sides. This concept is crucial for solving linear equations. Imagine a balanced scale, if you take the same weight from both sides, the scale remains balanced.

Let's consider the example equation \( z - 101 = -314 \). If we had started by subtracting 101 from both sides, it would have looked like this:
\[ (z - 101) - 101 = -314 - 101 \]

This approach ensures that the equation remains true. However, in our problem, the addition property is utilized. Both methods are complementary and serve to isolate the variable.
Addition Property of Equality
The Addition Property of Equality states that adding the same value to both sides of an equation keeps it balanced. This property is often used to simplify equations and solve for unknown variables.

In our example: \[ z - 101 = -314 \]
To isolate \( z \), we add 101 to both sides:
\[ (z - 101) + 101 = -314 + 101 \]
By doing this, we simplify it to:
\[ z = -213 \]
This shows that adding 101 to both sides does not change the equality, helping us get closer to finding the variable's value.
Isolating Variables
Isolating variables is the process of rearranging an equation to get the unknown variable on one side. This is a critical step in solving linear equations.

Let's break it down with our example:
We start with \[z - 101 = -314 \]
Our goal is to get \( z \) alone. We do this by using the Addition Property of Equality to move the constant term (101) from the left side.

Adding 101 to both sides gives us:
\[ z = -213 \]
This final step shows that when all other terms are moved to the other side, you are left with the isolated variable, which is the solution to the equation.
Understanding this concept is fundamental for more complex equations and mathematics in general.

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