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Chapter 3: Problem 1

Solve each equation. Check your solution. $$4 x=24$$

Short Answer

Expert verified
The solution is \(x = 6\), and substituting it back verifies the solution is correct.

Step by step solution

01

Understand the Equation

The given equation is a simple linear equation: \[4x = 24\]This means 4 times some number \(x\) equals 24.
02

Isolate the Variable

To solve for \(x\), you need to isolate it. Since \(x\) is being multiplied by 4, you do the opposite operation to both sides of the equation, which means dividing both sides by 4:\[\frac{4x}{4} = \frac{24}{4}\]This simplifies to:\[ x = 6 \]
03

Check the Solution

To ensure that the solution is correct, substitute \(x = 6\) back into the original equation:\[ 4 \times 6 = 24 \]Calculating the left side gives 24, which is equal to the right side of the equation. Therefore, the solution checks out.

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Key Concepts

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

Solving Linear Equations
Solving linear equations is a fundamental skill in algebra. It's about finding the value of a variable that makes the equation true. An equation is a mathematical statement that shows the equality of two expressions. In a linear equation like \(4x = 24\), each term is either a constant or a product of a constant and a variable.To solve an equation:
  • Identify the operation being performed on the variable.
  • Use the opposite operation to both sides of the equation to simplify.
  • Solve for the unknown variable.
In our problem, the equation \(4x = 24\) implies that the variable \(x\) is being multiplied by 4. The goal is to find what \(x\) should be to make the left side equal to the right side, which is 24.
Isolating Variables
Isolating the variable is a critical step when solving equations. It means getting the variable alone on one side of the equation. To isolate \(x\) in the equation \(4x = 24\), you'll perform arithmetic operations that undo whatever is being done to the variable.For our equation:
  • \(x\) is being multiplied by 4. To undo this, you should divide both sides of the equation by 4.
After dividing, the equation becomes \(x = 6\). This process ensures that you've found what value makes the equation true by leaving \(x\) alone on one side of the equation.
Checking Solutions
Checking your solutions is a crucial step in solving equations. You need to verify whether the value you've found truly satisfies the original equation. This helps prevent any errors you might have made during the calculations.To check the solution \(x = 6\) for \(4x = 24\):
  • Substitute \(x = 6\) back into the original equation: \(4 \times 6 = 24\).
  • Perform the multiplication: \(24 = 24\).
Since both sides of the equation are equal, the solution is correct. If the sides were not equal, it would indicate an error in solving or in the solution given.

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