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Chapter 2: Problem 23

Solve each equation using the Subtraction and Addition Properties of Equality. $$z+0.52=-8.5$$

Short Answer

Expert verified
The value of \( z \) is \( -9.02 \).

Step by step solution

01

Identify the Equation

The equation given is: \( z + 0.52 = -8.5 \)
02

Subtract 0.52 from Both Sides

To isolate \( z \), subtract 0.52 from both sides of the equation: \( z + 0.52 - 0.52 = -8.5 - 0.52 \)
03

Simplify Both Sides

Simplify the left side: \( z = -8.5 - 0.52 \)Then simplify the right side: \( z = -9.02 \)
04

Solution

The value of \( z \) is: \( z = -9.02 \)

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Key Concepts

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

Subtraction Property of Equality
To solve linear equations, one often needs to either add or subtract values from both sides of the equation. The Subtraction Property of Equality states that if you subtract the same number from both sides of an equation, the two sides remain equal.
Think of a balanced scale - if you take the same weight off both sides, it stays balanced.
Let's break down the concept with the given equation:
To isolate the variable, we need to eliminate the term that is combined with the variable. Here, we have:
  • Given equation: \[z + 0.52 = -8.5\]
  • To isolate \( z \), subtract 0.52 from both sides: \[z + 0.52 - 0.52 = -8.5 - 0.52\]
As you can see, subtracting the same amount from both sides keeps the equation balanced and helps simplify the equation by isolating the variable.
Addition Property of Equality
Just like the Subtraction Property of Equality, the Addition Property of Equality is another essential tool. This property states that if you add the same number to both sides of an equation, the equation remains balanced.
Consider the same balanced scale idea - if you add the same weight to both sides, the scale still balances.
In some cases, using the Addition Property of Equality is more appropriate. For instance, if the equation had a negative term combined with the variable, you would add to cancel it out:
  • If the equation was \[z - 0.52 = -8.5\], to isolate \( z \), add 0.52 to both sides: \[z - 0.52 + 0.52 = -8.5 + 0.52\]
  • Simplify to get: \[z = -8.5 + 0.52\]
By consistently applying the Addition Property of Equality, you can manipulate and solve linear equations in a straightforward manner.
Solving Linear Equations
Solving linear equations involves finding the value of the variable that makes the equation true. Linear equations are equations of the first degree, which means they have no exponents higher than 1. To solve such equations, follow these steps:
  • Identify the equation: Observe the given equation carefully and determine which operations are needed to isolate the variable. For example, given: \[ z + 0.52 = -8.5 \]
  • Isolate the variable: Use the Addition or Subtraction Property of Equality to eliminate constants on the same side as the variable. Here, subtract 0.52 from both sides: \[z = -8.5 - 0.52\]
  • Simplify both sides: Perform the arithmetic operation to simplify the equation and find the value of the variable. The simplified equation will be: \[ z = -9.02 \]

Practice applying these properties will make the process of solving linear equations second nature. Understanding and mastering these concepts are essential for success in algebra.

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