Problem 16 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 11 x=121 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 11 \).

Step 1: Identify the Equation

We are given the equation \( 11x = 121 \). Our task is to solve for \( x \).

Step 2: Isolate the Variable

To find \( x \), we need to isolate it. We do this by dividing both sides of the equation by 11: \[ x = \frac{121}{11} \].

Step 3: Simplify the Division

Now, perform the division on the right side of the equation: \[ x = 11 \].

Step 4: Verify the Solution

To ensure the solution is correct, substitute \( x = 11 \) back into the original equation. Calculate \( 11 \times 11 \) to check if it equals 121: \[ 11 \times 11 = 121 \]. Since both sides are equal, \( x = 11 \) is verified.

Key Concepts

Isolating the Variable

In the world of solving linear equations, one of the most crucial steps is to isolate the variable. This typically means getting the variable on one side of the equation while moving everything else to the opposite side. In the equation we have, \(11x = 121\), our primary focus is to find what \(x\) equals.

To isolate \(x\), we need to remove the coefficient (11 in this case) attached to it. We do this by performing the opposite operation. Since 11 is being multiplied with \(x\), we use division to cancel it out. Thus, dividing both sides by 11 gives you:

• Left side: \(11x / 11 = x\)
• Right side: \(121 / 11\)

The result is \(x = 11\). This is now a simplified equation where \(x\) stands alone, letting us clearly see its value.

Verifying the Solution

After obtaining a solution, it's vital to verify it. Verification is like a mathematical safety check to confirm you've got the correct answer. It involves substituting the solution back into the original equation and checking if both sides remain equal.

In our case, after isolating the variable, we found \(x = 11\). To verify, we substitute 11 back into the original equation, \(11x = 121\):

• Calculate: \(11 \times 11\)
• Result: \(121\)

Since both sides are equal, this confirms that our solution, \(x = 11\), is correct. Verification prevents errors and provides confidence in the accuracy of your solution.

Simplifying Division

Simplifying division is another fundamental aspect of solving equations, especially when dealing with coefficients. It ensures that we express fractions and ratios in their simplest form. In the context of our equation \(11x = 121\), after isolating the variable, we perform the division \(121 / 11\).

Breaking this down:

• The division itself is straightforward since 121 and 11 are both integers.
• 121 divided by 11 calculates directly to 11.

By simplifying, you not only confirm the value of \(x\) but also streamline the equation-solving process. It's critical to handle division carefully to avoid arithmetic errors and achieve an accurate solution.