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Chapter 8: Problem 5

In the following exercises, solve each equation. $$ x+7=12 $$

Short Answer

Expert verified
x = 5

Step by step solution

01

Identify the Equation

The given equation is: \[ x + 7 = 12 \]
02

Isolate the Variable

Subtract 7 from both sides of the equation to isolate the variable x. \[ x + 7 - 7 = 12 - 7 \] Simplify the left and right sides: \[ x = 5 \]
03

Verify the Solution

Substitute the value of x back into the original equation to verify: \[ 5 + 7 = 12 \] Since the left-hand side equals the right-hand side, the solution is correct.

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Key Concepts

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

Isolating the Variable
In solving linear equations, the goal is to find the value of the unknown variable. To achieve this, we need to get the variable by itself on one side of the equation. This process is known as isolating the variable. For our example, the equation is:
x + 7 = 12
To isolate the variable x, you need to perform an operation that will remove the constant number from the variable's side. Here, we subtract 7 from both sides:
x + 7 - 7 = 12 - 7
This simplifies to:
x = 5
This is our isolated variable, and we've found that x equals 5.
Verification of Solution
After isolating the variable and finding the solution, it's crucial to verify the solution. This step ensures that our solution is correct by substituting it back into the original equation. We need to check if both sides of the equation are equal.
Using the solution x = 5, substitute 5 back into the original equation:
5 + 7 = 12

Simplify the left side:
5 + 7 = 12
Since the left side equals the right side:
12 = 12

Our solution is verified and correct. This step is an essential part of solving equations to ensure the accuracy of our solution.
Basic Arithmetic Operations
Solving linear equations involves several basic arithmetic operations. These operations include addition, subtraction, multiplication, and division. Let's explore them briefly:
  • Addition: Combining two numbers to get a sum. For instance, adding 7 in the original equation.
  • Subtraction: Removing a number from another. In our example, subtracting 7 to isolate x.
  • Multiplication: Increasing a number by a factor. Not directly used in our example, but may be required for other equations.
  • Division: Splitting a number into equal parts. Also not directly used in our example, but can be key in other problems.

To isolate the variable efficiently, understanding and applying these operations correctly is essential. For our given problem, subtraction was the operation needed to move the constant term across the equals sign, helping us isolate x. Familiarity with these operations will make solving equations a smoother process.
Understanding when and how to use each basic arithmetic operation is a cornerstone of mastering algebra.

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