Problem 27 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 3 m-1=-13 $$

Step-by-Step Solution

Answer

The solution is \(m = -4\).

Step 1: Isolate the Variable Term

To solve the equation, start by isolating the term containing the variable \(m\). The current equation is \(3m - 1 = -13\). Add 1 to both sides to get rid of the -1: \(3m - 1 + 1 = -13 + 1\). This simplifies to \(3m = -12\).

Step 2: Solve for the Variable

Now, divide both sides of the equation by 3 to solve for \(m\): \(\frac{3m}{3} = \frac{-12}{3}\). This simplifies to \(m = -4\).

Step 3: Check the Solution

Substitute \(m = -4\) back into the original equation to verify your solution: \(3(-4) - 1 = -13\). Simplify the left-hand side: \(-12 - 1 = -13\). Since both sides of the equation are equal, the solution \(m = -4\) is correct.

Key Concepts

Isolating the Variable

When solving linear equations, one of the first steps is often isolating the variable. This means you want to get the unknown, like our variable \( m \), by itself on one side of the equation.
In the original equation \( 3m - 1 = -13 \), our goal is to have \( m \) all alone on one side.

• First, look at the terms around \( m \). There is a \(-1\) on the left side that is subtracting from \( m \).
• To eliminate this \(-1\), we add 1 to both sides of the equation. This operation cancels the \(-1\) and gives us \( 3m = -12 \).

By isolating the variable, you set yourself up to solve the equation more straightforwardly. Remember, every action you do to one side, you must do to the other to maintain the equation's balance.

Checking Solutions

After finding the potential solution of an equation, checking it is an important verification step. This helps confirm that your solution is indeed correct.
Let's use our equation again: \( 3m - 1 = -13 \) and the solution we found \( m = -4 \).

• First, substitute back the solution into the original equation. So it turns into \( 3(-4) - 1 = -13 \).
• Calculate the left-hand side: \( 3 \times (-4) = -12 \).
• Now subtract 1 to get \(-12 - 1 = -13 \).

Both sides of the equation are equal, so our solution \( m = -4 \) checks out!
Checking solutions ensures that your calculations are correct and avoids errors.

Algebraic Manipulation

Algebraic manipulation is a key skill in solving equations. It involves rearranging and simplifying equations to make them more workable.
In our equation, \( 3m - 1 = -13 \), manipulation plays a crucial role in both isolating the variable and solving for it.

• First, we removed \(-1\) by adding 1 to both sides, using basic operations to change the equation's layout.
• Next, we needed to solve for \( m \). This required dividing both sides by 3, another form of manipulation.

These steps illustrate the power of algebraic manipulation:
- You can use addition or subtraction to clear constants.- Multiplication or division helps isolate variables or simplify equations further.Algebraic manipulation is the toolbox you will reach for time and again in mathematics.