Problem 31 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ \frac{m}{7}=-8 $$

Step-by-Step Solution

Answer

Answer: The value of \(m\) in the given equation is \(m = -56\).

Step 1: Identify the operation on the variable m

In the given equation, \(\frac{m}{7}=-8\), the variable \(m\) is divided by 7.

Step 2: Perform the inverse operation

The inverse operation of division is multiplication. In order to isolate \(m\), we will multiply both sides of the equation by 7. $$ 7\cdot\frac{m}{7} = 7\cdot(-8) $$

Step 3: Simplify both sides of the equation

The 7's on the left will cancel out, leaving only the variable \(m\): $$ m=-8\cdot7 $$ Now, we can perform the multiplication on the right side of the equation: $$ m=-56 $$

Step 4: Interpret the result

The solution to the equation \(\frac{m}{7} = -8\) is \(m = -56\). This means that when \(m\) equals \(-56\), the equation is true.

Key Concepts

Inverse Operations

Understanding inverse operations is crucial when solving algebraic equations. It's how we reverse the action of an operation in order to solve for a variable. Think of it as doing the opposite: if you're given a multiplication, you'll divide; if you're given a division, you'll multiply; and the same goes for addition and subtraction. In the exercise \(\frac{m}{7}=-8\), the variable \(m\) is divided by 7. To reverse this, you multiply both sides by 7. This is because multiplication and division are inverses of each other. Here's the visual breakdown:

\[7 \cdot \frac{m}{7} = 7 \cdot (-8)\]
After performing the inverse operation, the equation becomes simpler, and the variable \(m\) is one step closer to being isolated.

Isolating Variables

Isolating variables means getting the variable you're solving for, in this case \(m\), by itself on one side of the equation. This is the heart of solving an equation: manipulating the equation to have the variable stand alone. After using inverse operations, you're at the stage where the variable is almost isolated, but you may need to further simplify the equation.

After multiplying both sides by 7, you get \(m\) on its own on the left:
\[m = -56\]
This informs you that the value of \(m\) that makes the original equation true is \(m = -56\). Isolating variables is like a treasure hunt where each step gets you closer to the 'X' that marks the spot!

Simplifying Equations

Simplifying equations is the process of making them easier to understand and solve. This often involves combining like terms, reducing fractions, or removing parentheses. In our example, simplifying comes down to the basic arithmetic of calculating \(7 \cdot (-8)\). It's straightforward but crucial. Simplifying is all about cleaning up the equation to make the solution clear:

\[m = -56\]
At this point, the equation is simplified enough that you can easily see the solution. Simplifying is an essential skill that helps prevent mistakes and can make complex equations look a lot less intimidating.