Problem 121 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 3 m+4=2 m+1 $$

Step-by-Step Solution

Answer

The solution to the equation is \( m = -3 \).

Step 1: Understand the Equation

The given equation is \( 3m + 4 = 2m + 1 \). This is a linear equation where we need to find the value of \( m \) that satisfies the equation.

Step 2: Isolate the Variable Terms

First, we need to get all terms with \( m \) on one side. Subtract \( 2m \) from both sides of the equation:\[3m + 4 - 2m = 2m + 1 - 2m\]This simplifies to:\[m + 4 = 1\]

Step 3: Simplify the Constant Terms

Next, we want to isolate \( m \) by removing the constant on the left side. Subtract 4 from both sides of the equation:\[m + 4 - 4 = 1 - 4\]This simplifies to:\[m = -3\]

Step 4: Verify the Solution

Substitute \( m = -3 \) back into the original equation to verify correctness:\[3(-3) + 4 = 2(-3) + 1\]This simplifies to:\[-9 + 4 = -6 + 1;\ -5 = -5\]Since both sides of the equation are equal, the solution \( m = -3 \) is correct.

Key Concepts

Solving Equations

Solving linear equations may seem daunting at first, but it boils down to finding a number that makes the equation true. The process involves manipulating the equation while ensuring that the equality stays balanced. This means whatever operation is done to one side must equally be done to the other side.

In our example, the equation is given as \( 3m + 4 = 2m + 1 \). We are expected to find the value of \( m \) that would make both sides equal. By carefully following the steps, equations can be solved systematically, turning what seems complex into simple math. Solving equations is not just about finding \( m \), but also about understanding the logic of balancing scales, where both sides remain equal throughout the operations.

Variable Isolation

Variable isolation is a key step in solving equations, where we rearrange the equation to get the variable, in this case, \( m \), by itself on one side of the equation.

To isolate \( m \) from the equation \( 3m + 4 = 2m + 1 \), we start by removing the term containing \( m \) from the right side. We do this by subtracting \( 2m \) from both sides, which simplifies the equation to \( m + 4 = 1 \). Here, \( m \) is closer to being isolated but is still attached to the constant +4.

We do another operation to completely isolate \( m \); subtracting 4 from both sides gives us \( m = -3 \). Now we have isolated \( m \), making it straightforward to see our solution.

Equation Verification

Once a solution is found, equation verification ensures that our solution is indeed correct. This involves substituting our found solution back into the original equation and checking if both sides of the equation are equal.

In our problem, after finding \( m = -3 \), we plug it back into the original equation: \( 3(-3) + 4 = 2(-3) + 1 \). Simplifying each side gives \(-5 = -5\), confirming our solution is accurate.

Verification is crucial as it helps us catch any potential mistakes made during the solving process. By performing this step, we ensure that the solution is not just mathematically valid but also the correct answer to the problem posed.