Problem 23 - Elementary Algebra

Question

Solve the equations. $$ \frac{m}{6}+4=8 $$

Step-by-Step Solution

Answer

Equation: $$\frac{m}{6} + 4 = 8$$ Answer: The value of m is 24.

Step 1: Isolate the term with the variable

Subtract 4 from both sides of the equation to isolate the term with the variable (m/6) on one side: $$ \frac{m}{6} = 8 - 4 $$

Step 2: Clear the fraction

Multiply both sides of the equation by 6 (the denominator) to clear the fraction: $$ 6 \times \frac{m}{6} = 4 \times 6 $$ This simplifies to: $$ m = 24 $$

Step 3: Check the solution

Substitute the value of m (24) back into the original equation to check if it's correct: $$ \frac{24}{6} + 4 = 8 $$ Simplify the left side of the equation: $$ 4 + 4 = 8 $$ As 8 = 8, the solution is correct. Therefore, the value of m is 24.

Key Concepts

Isolation of Variable

When solving linear equations, the goal is to find the value of the variable that makes the equation true. One of the fundamental steps in this process is the isolation of the variable. In our example, we have the equation \( \frac{m}{6} + 4 = 8 \). To isolate the variable term, \( \frac{m}{6} \), on one side of the equation, we need to remove the constant term from that side. This is done by performing the opposite operation. Since 4 is being added to \( \frac{m}{6} \), we subtract 4 from both sides of the equation:

• Left side after subtraction: \( \frac{m}{6} = \)
• Right side becomes: \( 8 - 4 \)

Thus, our equation simplifies to \( \frac{m}{6} = 4 \). This isolation of the variable term makes it easier for us to solve for \( m \) in the subsequent steps. Remember, whatever you do to one side of the equation, do to the other. This balance is crucial to maintain equality.

Clearing Fractions

Dealing with fractions in equations can be daunting, but there's a handy technique to simplify your calculations: clearing fractions. This involves eliminating the fraction by multiplying both sides of the equation by the same number. In our exercise, we have \( \frac{m}{6} = 4 \). The fraction \( \frac{m}{6} \) can be cleared by multiplying both sides of the equation by 6, which is the denominator of the fraction.

• Multiply left side: \( 6 \times \frac{m}{6} \)
• Multiply right side: \( 4 \times 6 \)

The left side simplifies as \( m \) (since \( 6 \times \frac{m}{6} = m \)) and the right side as \( 24 \), leading us to the equation \( m = 24 \). This step effectively removes fractions from the picture, making it a breeze to solve for the variable.

Checking Solutions

After solving an equation, it's important to verify that the solution is correct. This is where checking solutions comes into play. In our example, we've determined that \( m = 24 \). To check this solution, substitute \( m = 24 \) back into the original equation: \( \frac{24}{6} + 4 = 8 \).

• Simplify the left side: \( \frac{24}{6} = 4 \)
• Add 4 to this: \( 4 + 4 \)
• The result should equal the right side of the equation: \( 8 \)

Indeed, \( 4 + 4 = 8 \), confirming the correctness of our solution. This final check is a critical step in problem-solving, as it ensures there were no errors in our calculations, providing confidence in the solution found.