Problem 51 - Prealgebra

Question

Solve each equation. Check your solution $$108=18 x$$

Step-by-Step Solution

Answer

The solution is \(x = 6\), and it is verified by substitution.

Step 1: Isolate the Variable

To solve the equation \(108 = 18x\), we need to isolate the variable \(x\). We can do this by dividing both sides of the equation by 18, which is the coefficient of \(x\). After performing the division, we get: \(x = \frac{108}{18}\).

Step 2: Simplify the Equation

Now, simplify the expression \(\frac{108}{18}\). This can be simplified by dividing 108 by 18. The division gives us a result of 6. Thus, \(x = 6\).

Step 3: Verify the Solution

To check our solution, substitute \(x = 6\) back into the original equation \(108 = 18x\). Substitute 6 for \(x\) to get \(108 = 18\times 6\). Calculate the right-hand side of the equation: \(18 \times 6 = 108\). Since both sides are equal, \(x = 6\) is verified as correct.

Key Concepts

Isolation of Variable

Isolating the variable is a crucial step in solving equations. It means getting the variable on its own on one side of the equation. In our example, the variable is \(x\) in the equation \(108 = 18x\). To isolate \(x\), we need to eliminate any numbers that are multiplied with it.

• Identify the coefficient: In our case, it is 18, as indicated by \(18x\). It's the number directly multiplying \(x\).
• Move the coefficient: We achieve this by using the opposite operation to what's being done to the variable. Since \(x\) is being multiplied by 18, we divide by 18 on both sides to "cancel it out."
  
  Performing the division step: \( 108 = 18x \rightarrow x = \frac{108}{18}\)

Consequently, \(x\) is separated and the equation looks cleaner. The variable is now alone on one side, paving the way for further simplification.

Division Method

The division method is often used to solve equations where the variable is multiplied by a coefficient. Here, dividing both sides of the equation by the same number keeps the equation balanced, a fundamental rule in algebra. In the example \(108 = 18x\), we have successfully isolated \(x\) and

• Divide to simplify: We divided both sides by 18, which is the coefficient of \(x\). As a result, we got \(x = \frac{108}{18}\).
• Divide further: Calculate \(\frac{108}{18}\), which equals 6. So, \(x = 6\).

This shows that the division method not only isolates the variable but also simplifies the expression to its core solution. It turns complex-looking equations into simple, understandable ones.

Verification of Solution

Verification is the step where you check if your solution actually satisfies the original equation. This process assures us the solution is correct. With \(x = 6\), substitute it back into the original equation \(108 = 18x\) to see if it holds true:

• Substitute back: Replace \(x\) with 6 in the equation, so it becomes \(108 = 18 \times 6\).
• Calculate the right side: \(18 \times 6\) results in \(108\).

Since both sides match, \(x = 6\) is a valid solution. Verification validates our steps and assures that no mistakes were made during calculations, confirming that our solution is indeed accurate.