Problem 20 - Algebra 1

Question

Solve the equation if possible. $$ -2 m=16 m-9 $$

Step-by-Step Solution

Answer

The solution to the equation -2m = 16m - 9 is \( m = \frac{1}{2} \)

Step 1: Simplify the equation by getting all terms with m on one side

Add 2m on both sides of the equation: \[ -2m + 2m = 16m - 9 + 2m \], which leads to \[ 0 = 18m - 9 \]

Step 2: Isolate the term with the variable m

Add 9 on both sides of the equation: \[ 0 + 9 = 18m - 9 + 9 \], leading to: \[ 9 = 18m \]

Step 3: Solve for m

Divide both sides by 18: \[ \frac{9}{18} = \frac{18m}{18} \] leading to \[ m = \frac{1}{2} \]

Key Concepts

Equation Simplification

The first critical step in solving linear equations is simplification. This involves rearranging the equation to combine like terms, which makes further steps easier. In the given problem, we begin with the equation \(-2m = 16m - 9\).

Simplification in this context involves moving all the terms involving the variable to one side.

• We want to eliminate \(-2m\) from the left side by adding \(2m\) to both sides.
• This results in the new equation \(0 = 18m - 9\).

By doing this, we have successfully simplified the equation. This step helps set the stage for isolating the variable, which will ultimately allow us to solve the equation.

Isolating the Variable

Once the equation is simplified, the next step is to isolate the variable—usually the letter (often 'm' or 'x') that we're solving for. In our example, after simplification, we have the equation \(0 = 18m - 9\).
This equation suggests that we're one step away from isolating \(m\).

To isolate \(m\), we need to eliminate any constant terms on the side where \(m\) resides.

• In our case, we can add \(9\) to both sides of the equation: \(0 + 9 = 18m - 9 + 9\).
• This subtraction and addition of \(9\) simplifies the equation to \(9 = 18m\).

Now, all terms containing \(m\) are isolated on one side of the equation, making it clear and easy to solve for \(m\). This clarity allows us to focus on only one last step—solving for the variable.

Arithmetic Operations

The final step in solving the equation involves basic arithmetic operations. At this stage of our problem, we have the isolated equation \(9 = 18m\). This points to a straightforward arithmetic operation: division.

• Divide each side of the equation by \(18\) to solve for \(m\).
• This division looks like \(\frac{9}{18} = \frac{18m}{18}\).

Performing this division simplifies the equation to \(m = \frac{1}{2}\). Using arithmetic operations such as addition, subtraction, multiplication, and division is essential for manipulating equations to solve for unknown variables.

Each operation should ensure the equation remains balanced, just like numbers on a scale, leading us to the correct solution, which in this case is \(m = \frac{1}{2}\).