Problem 31 - Prealgebra

Question

Solve each equation. Check your solution. $$-68=-4 m$$

Step-by-Step Solution

Answer

The solution is \(m = 17\).

Step 1: Identify the Equation

The given equation is \(-68 = -4m\). Our goal is to solve for \(m\).

Step 2: Isolate the Variable

To solve for \(m\), divide both sides of the equation by \(-4\) to isolate \(m\). This gives: \(\frac{-68}{-4} = m\).

Step 3: Simplify the Expression

Calculate \(\frac{-68}{-4}\), which results in \(17\). Therefore, \(m = 17\).

Step 4: Check the Solution

Substitute \(m = 17\) back into the original equation: \(-68 = -4(17)\). Simplify the right side: \(-4 \times 17 = -68\). Both sides are equal, confirming our solution.

Key Concepts

Isolating Variables

One of the fundamental steps in solving linear equations is isolating the variable. This means getting the variable alone on one side of the equation. Think of it like a game: your goal is to find what the variable equals without any other numbers or operations on its side. In our exercise, the equation is \(-68 = -4m\). To isolate \(m\), we need to eliminate the coefficient \(-4\) from \(m\). We can do this by performing the inverse operation. Because \(-4\) is multiplied by \(m\), we divide both sides by \(-4\). This operation is: \[\frac{-68}{-4} = m\]. After this division, we successfully isolate \(m\), making it easier to solve what \(m\) really is.

Checking Solutions

Once we find a potential solution, it is crucial to check if it's correct. This ensures our steps were accurate and that we didn't make a mistake along the way. In mathematics, checking solutions is like a quality control step. For our equation, after solving \(m = 17\), we substitute \(m\) back into the original equation: \(-68 = -4(17)\). Calculate the right-hand side: \(-4 \times 17 = -68\). Since both sides of the equation match, it confirms that our solution \(m = 17\) is correct. If they didn't match, it would mean we'd have to recheck our steps or calculations.

Simplifying Expressions

Simplifying expressions is about making calculations easier and removing any unnecessary complexity. In linear equations, this often involves arithmetic simplifications. In our given problem, after isolating \(m\), the expression was \(\frac{-68}{-4}\). The simplification requires us to divide the numbers. Performing this calculation, \(\frac{-68}{-4} = 17\), yields a simpler, single number \(17\) for \(m\). This step reduces any complication in future calculations and gives us a cleaner, more understandable result. Simplifying is crucial for getting precise and easily verifiable answers.

Algebra Basics

Algebra basics lay the groundwork for solving equations. These fundamentals include understanding operations (addition, subtraction, multiplication, division), equations, and balancing both sides of any mathematical statement. Consider the equation \(-68 = -4m\). In algebra, every operation has an inverse which helps in solving equations. Understanding how and when to use these inverse operations is part of algebra basics. For instance, to get rid of multiplication by \(-4\), we use division. Another key concept is maintaining balance. Every operation you perform on one side must be performed on the other. This balance is what forms the core of algebra, ensuring all steps lead to a valid solution. Mastering these basics helps in tackling more complex problems with confidence.