Problem 21 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 2 m=-62 $$

Step-by-Step Solution

Answer

The solution is \(m = -31\).

Step 1: Isolate the Variable

We're given the equation \(2m = -62\). To solve for \(m\), we need to isolate it. Start by dividing both sides of the equation by 2: \[ m = \frac{-62}{2} \]

Step 2: Simplify the Expression

Simplify the right side of the equation. Dividing -62 by 2 gives us: \[ m = -31 \]

Step 3: Verify the Solution

To ensure that our solution \(m = -31\) is correct, substitute \(m\) back into the original equation: \[ 2(-31) = -62 \] Simplify the left side: \(-62 = -62\). Since both sides of the equation are equal, our solution is verified.

Key Concepts

Isolating the Variable

In order to solve any linear equation, one of the first things you need to do is isolate the variable. This means getting the variable by itself on one side of the equation, usually the left side. For example, in the equation \(2m = -62\), the variable is \(m\). It's currently multiplied by 2. To isolate \(m\), you need to perform the opposite operation of multiplication, which is division.
Divide both sides of the equation by 2 to keep it balanced:

• \(\frac{2m}{2} = \frac{-62}{2}\)
• This simplifies to \(m = -31\)

This process of isolating makes it easy to figure out what the variable equals. Remember, whatever you do to one side of the equation, you must do to the other side to maintain equality.

Simplifying Expressions

Simplifying expressions is another crucial step when solving equations. After isolating the variable, you often have to simplify what is left. In our example, once you divide both sides by 2, you arrive at the expression \(m = \frac{-62}{2}\).
Now, simplify by performing the division:

• Calculate \(\frac{-62}{2} = -31\)

This step is all about breaking down the expression to its simplest form, which makes it clear and easy to understand. You should always simplify expressions as much as possible to find the most straightforward solution.

Verifying Solutions

Verifying your solutions is an essential part of solving equations, especially in a classroom or homework setting. This process assures you that your solution is correct. To verify, you substitute the found solution back into the original equation and check whether the equation holds true.
For example, substitute \(m = -31\) back into the original equation \(2m = -62\):

• Calculate \(2(-31) = -62\)
• Simplify the left side to \(-62\)

Since the left side equals the right side, \(-62 = -62\), the solution \(m = -31\) is verified and correct. Always make sure that both sides of the equation equal after substitution; this confirms the accuracy of your solution.