Problem 22 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 3 m=-54 $$

Step-by-Step Solution

Answer

The solution is \(m = -18\).

Step 1: Understand the Problem

The equation given is \(3m = -54\). Our goal is to find the value of \(m\) that makes this equation true.

Step 2: Isolate the Variable

To solve for \(m\), you need to get \(m\) by itself on one side of the equation. Right now, \(m\) is being multiplied by 3, so you will divide both sides of the equation by 3 to cancel out the multiplication. Divide both sides by 3 to get \(m = \frac{-54}{3}\).

Step 3: Simplification

Calculate \(-54 \div 3\). Dividing \(-54\) by 3 gives \(-18\). So, \(m = -18\).

Step 4: Verification

Substitute \(-18\) back into the original equation to verify the solution: \(3(-18) = -54\). Calculate the left side: \(3 \times -18 = -54\), which is equal to the right side of the equation. Therefore, the solution is verified.

Key Concepts

Isolation of Variables

When solving linear equations, the first major step often involves isolating the variable. In our exercise, the variable is \(m\) in the equation \(3m = -54\). To isolate \(m\), we want it to be by itself on one side of the equation. This means we need to remove the number multiplying it, which is 3 in this case.
To achieve isolation, we perform the opposite operation. Since \(m\) is being multiplied by 3, we do the opposite, which is division, to both sides. Divide both sides by 3. This gives us:

• \(m = \frac{-54}{3}\)

By dividing, we've successfully isolated \(m\). This step is crucial in solving any equation because it simplifies the problem to finding out the value of the variable on its own.
Remember, whatever operation you do to one side of the equation, you must apply to the other side as well. This keeps the equation balanced and ensures that no changes have been made to the equality of the statement.

Verification of Solutions

Verification is a critical step in solving mathematical equations and is often overlooked, yet it confirms the accuracy of your solution. Once we obtained \(m = -18\), it's essential to check that this value satisfies the original equation. To verify, substitute \(-18\) back into the original equation \(3m = -54\):
\(3(-18) = -54\)
Calculate the left side:

• \(3 \times -18 = -54\)

The left side equals the right side of the original equation, \(-54 = -54\).
This consistency verifies our solution is correct. Verification not only reassures the solver but also helps in revealing any arithmetic errors made during calculations.

Simplification in Algebra

Simplification plays a key role in algebra, making your results easier to interpret and work with. After isolating \(m\) into \(m = \frac{-54}{3}\), you simplify the fraction.
Perform the division:

• \(-54 \div 3 = -18\)

Simplification means performing all possible operations until the equation or expression cannot be reduced any further.
This step is especially important for presenting clear results in algebraic problems. Simpler expressions are easier to verify and use in further calculations or applications. In this case, reducing \(m\) to \(-18\) gives us the simplest form of the solution to work with in any subsequent steps.