Problem 44 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ \frac{2 m}{9}=4 $$

Step-by-Step Solution

Answer

The solution is \( m = 18 \).

Step 1: Isolate the Variable

To solve the equation \( \frac{2m}{9} = 4 \), start by isolating \( m \) on one side of the equation. Multiply both sides of the equation by 9 to eliminate the fraction. This gives:\[ 2m = 36\]

Step 2: Solve for the Variable

Next, solve for \( m \) by dividing both sides of the equation by 2:\[m = \frac{36}{2}\]Simplifying the right side, you get:\[m = 18\]

Step 3: Verify the Solution

To ensure the solution is correct, substitute \( m = 18 \) back into the original equation:\[\frac{2(18)}{9} = 4\]Simplify the left-hand side:\[\frac{36}{9} = 4\]Which confirms:\[4 = 4\]Thus, the solution \( m = 18 \) is correct.

Key Concepts

Isolating Variables

One of the first steps in solving linear equations is isolating the variable, which means getting the variable of interest by itself on one side of the equation. This involves reversing the operations that are being performed on the variable. Think of it as peeling back layers to reveal what's underneath, which, in this case, is the variable itself.
To isolate a variable:

• Identify the operations applied to the variable.
• Perform inverse operations to undo these operations.
• Whatever you do to one side of the equation, you must do to the other side to maintain equality.

In our example, the goal was to isolate \( m \) in the fraction \( \frac{2m}{9} = 4 \). We multiplied both sides by 9 to remove the fraction, leaving us with \( 2m = 36 \). This step is crucial because it simplifies the equation into a form that's easier to solve.

Fractions in Equations

Dealing with fractions in equations can be intimidating, but it doesn't have to be. When fractions are involved, a common technique is to eliminate the fractions by multiplying both sides of the equation by a common denominator. This transforms the problem into a simpler, fraction-free equation.
The process of removing fractions:

• Identify the denominator in the fraction.
• Multiply both sides of the equation by this denominator to clear the fraction.
• Simplify the resulting equation.

In the equation \( \frac{2m}{9} = 4 \), multiplying both sides by 9 effectively cleared the fraction, making it much easier to continue solving for \( m \). Now with \( 2m = 36 \), we have a fraction-free equation that's straightforward to tackle.

Checking Solutions

Always check your solutions to ensure accuracy and understanding. Verifying the solution involves substituting your calculated value back into the original equation and ensuring the equality holds true. This step confirms that you have not made a mistake and reinforces your confidence in the solution.
How to check your solution:

• Take the solution you've found and substitute it back into the original equation.
• Simplify both sides of the equation to check if they are equal.
• If they are, your solution is verified.

In this case, substituting \( m = 18 \) back into the original equation \( \frac{2(18)}{9} = 4 \) simplifies to \( \frac{36}{9} = 4 \), proving that \( 4 = 4 \). This confirms the solution \( m = 18 \) is indeed correct.