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Chapter 1: Problem 11

In Problems 11-18, mentally solve each equation. $$ 7 x=21 $$

Short Answer

Expert verified
x = 3

Step by step solution

01

Understand the Equation

The given equation is a simple linear equation: \(7x = 21\). It means 7 times a number (x) equals 21. Our task is to find the value of x.
02

Isolate the Variable

To find the value of x, divide both sides of the equation by 7. This will isolate x on one side of the equation.
03

Perform the Division

When you divide both sides by 7, you get \( x = \frac{21}{7} \).
04

Simplify the Right Side

Divide 21 by 7 to simplify the equation: \( \frac{21}{7} = 3 \). So, \( x = 3 \).
05

Verify the Solution

To check if the solution is correct, substitute x back into the original equation: \( 7 \times 3 = 21 \). Since both sides equal 21, the solution is verified to be correct.

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Key Concepts

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

Solving Equations
Solving equations is a crucial skill in mathematics. It involves finding the value of the unknown variable that makes the equation true. In our example, we have the equation: 7x = 21. The goal is to determine the value of x. Solving equations often starts with understanding the given problem and determining the steps needed to isolate the variable.
This process usually includes operations such as addition, subtraction, multiplication, and division.
In linear equations like this one, solving becomes straightforward with the right approach, making it easy to find x.
Isolate the Variable
Isolating the variable means getting the unknown variable (x) by itself on one side of the equation. This is often done through inverse operations.
For the equation: 7x = 21, we need to isolate x. Here, x is multiplied by 7, so we use the inverse operation, which is division.
We divide both sides of the equation by 7 to isolate x.
The steps are as follows:
  • Original equation: 7x = 21
  • Divide both sides by 7: \( \frac{7x}{7} = \frac{21}{7} \)
  • This simplifies to: x = 3
By doing this, we have isolated the variable x.
Simplification
Simplification is key in solving equations. It's about making the equation as simple as possible. In our problem, after isolating x by dividing both sides by 7, we have the fraction \( \frac{21}{7} \).
Simplifying this fraction is straightforward: 21 divided by 7 equals 3. Therefore, \( x = 3 \).
Simplification helps us find the most reduced form of our solution, making it easier to understand and verify.
Verify Solution
Verifying the solution is an essential step to ensure the accuracy of our work. After finding that \( x = 3 \), we substitute x back into the original equation to check if it holds true.
Here's the verification:
Start with the original equation: 7x = 21. Substitute x with 3: 7 \( \times \) 3.Calculating this, we get: 21 = 21.
Since both sides of the equation are equal, our solution \( x = 3 \) is correct. This step gives us confidence that we've solved the equation accurately.

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