Problem 102 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 4 x=24 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 6\).

Step 1: Identify the Equation

We are given the equation \(4x = 24\). This is a straightforward linear equation.

Step 2: Isolate the Variable

To solve for \(x\), we need to isolate it on one side of the equation. This can be done by dividing both sides of the equation by 4, since 4 is currently being multiplied by \(x\).

Step 3: Perform the Division

Divide both sides of the equation by 4: \(\frac{4x}{4} = \frac{24}{4}\). This simplifies to \(x = 6\).

Step 4: Verify the Solution

Double-check your solution by substituting \(x = 6\) back into the original equation. Calculate: \(4 \times 6 = 24\). Since both sides are equal, the solution is verified.

Key Concepts

Isolating the variable

When solving equations, our main goal is to find the value of the variable that makes the equation true. To do this, it's essential to have the variable on one side of the equation by itself. This process is known as "isolating the variable." In the given equation, \(4x = 24\), the variable \(x\) is multiplied by 4. To isolate \(x\), we must remove the multiplication by 4.
- Multiplication is reversed by division, which means that we need to divide both sides of the equation by 4.- This step is critical as it allows us to express \(x\) alone on one side, making it possible to solve for its value.Remember, whatever operation you perform on one side of the equation, you must also perform on the other side. This maintains the balance of the equation, ensuring that both sides remain equal. Isolating the variable is the first crucial step towards finding the solution.

Division in equations

Once we've decided to isolate the variable, the next step involves using division to remove the coefficient from the variable. In the equation \(4x = 24\), 4 is the coefficient of \(x\), and to eliminate it, we divide every part of the equation by 4.
Here's how it works:

• Write \[\frac{4x}{4} = \frac{24}{4}\]
• This leads to \(x = 6\), as the 4s cancel each other out on the left side.

This is possible because of the division property of equations, which states that dividing both sides of an equation by the same nonzero number will not change the equation's solutions. It is a powerful tool that helps pivot from a more complex appearance to a simplified solution.

Verification of solutions

Verifying your solution is an essential step to ensure that it is correct. In this case, once we determined that \(x = 6\), we should substitute \(x\) back into the original equation, \(4x = 24\), to check our work.
Here's how you verify:

• Substitute \(x = 6\) back into the equation: \(4 \times 6 = 24\).
• Perform the calculation: 24 does indeed equal 24.

Since both sides of the equation are identical after the substitution, it confirms our solution is correct. Verification of solutions is the final and crucial check in solving equations, acting as a safeguard against errors in previous calculations. This ensures confidence in the accuracy of your solution.