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Chapter 2: Problem 5

Solve each equation using the Subtraction and Addition Properties of Equality. $$x+24=35$$

Short Answer

Expert verified
x = 11

Step by step solution

01

Identify the Goal

The goal is to isolate the variable, which means getting the variable on one side of the equation by itself.
02

Subtraction Property of Equality

To isolate the variable, subtract 24 from both sides of the equation: \(x + 24 - 24 = 35 - 24\).
03

Simplify Both Sides

Simplify each side of the equation: \(x + 0 = 11\).
04

Final Solution

After simplifying, the equation is now \(x = 11\). This is the solution.

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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 is a fundamental principle in algebra. It states that you can subtract the same number from both sides of an equation without changing the equality. This is a powerful tool for isolating variables. For example, in the equation \(x + 24 = 35\), we want to get \(x\) by itself on one side. To do this, we subtract 24 from both sides:

  • Original equation: \(x + 24 = 35\)
  • Subtract 24 from both sides: \(x + 24 - 24 = 35 - 24\)
  • This simplifies to: \(x = 11\)
By applying the Subtraction Property of Equality, we maintain a balanced equation, leading us to the correct solution.
Isolate the Variable
Isolating the variable means getting the variable alone on one side of the equation. The primary goal is to find out what the variable equals by removing any other numbers or terms from its side.


Steps to isolate the variable:
  • Identify the term containing the variable.
  • Use properties of equality, such as addition, subtraction, multiplication, or division, to remove other terms.
In the case of \(x + 24 = 35\), the term with the variable is \(x\). To isolate \(x\), we subtract 24 from both sides using the Subtraction Property of Equality: \(x + 24 - 24 = 35 - 24\). This leaves us with \(x = 11\). Now, \(x\) is isolated, and we have the solution.
Simplifying Equations
Simplifying equations involves performing operations to make the equation easier to solve. After applying properties of equality, ensure each term is fully simplified. This can involve combining like terms, reducing fractions, or performing basic arithmetic.

In our example, after subtracting 24 from both sides of \(x + 24 = 35\), we get \(x = 11\). Here, we simplify the left side: \(x + 24 - 24\) simplifies to \(x\) because \(24 - 24 = 0\). The right side simplifies to 11 because \(35 - 24 = 11\).

Steps for simplifying equations:
  • Perform arithmetic operations like addition or subtraction.
  • Combine like terms to simplify further.
  • Check both sides of the equation to make sure they are fully simplified.
Through these steps, the equation becomes easier to understand and solve, leading you to the correct answer.

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