Problem 63 - Prealgebra

Question

Solve each equation. Check your solution. $$25 m=225$$

Step-by-Step Solution

Answer

The solution is \(m = 9\). This value satisfies the original equation.

Step 1: Identify the Equation

We start with the equation given in the problem: \(25m = 225\). The goal is to solve for \(m\).

Step 2: Isolate the Variable

To get \(m\) by itself, we need to divide both sides of the equation by 25, the coefficient of \(m\). So, we have: \[m = \frac{225}{25}\].

Step 3: Perform the Division

Calculate \(\frac{225}{25}\). By dividing 225 by 25, we find that:\[m = 9\].

Step 4: Check the Solution

Substitute \(m = 9\) back into the original equation to verify it works: - Compute: \(25 \times 9 = 225\). - Since both sides are equal \(225 = 225\), the solution checks out.

Key Concepts

Isolating Variables

When solving equations, isolating the variable is an essential step. It involves rearranging the equation in such a way that the unknown variable appears on one side of the equation all by itself. This process allows you to determine the value of the variable.

To isolate the variable, identify the operations surrounding the variable. Consider them as obstacles that need clearing. You should do the opposite operation to "cancel out" these obstacles. For instance, if a variable is multiplied by a number, like in the equation \(25m = 225\), the solution requires dividing both sides by that number (25 in this case). This division "undoes" the multiplication, leaving you with just the variable \(m\) on one side, as shown:

• Equation: \(25m = 225\)
• Operation: Divide by 25
• Result: \(m = \frac{225}{25}\)

Isolating variables helps simplify the process and sets the stage for finding a clear numerical solution.

Division in Equations

Division is often used as a key operation when solving equations, particularly those involving a multiplication with a variable. By using division, we can simplify an equation to find the value of an unknown variable.

In the equation \(25m = 225\), the variable \(m\) is multiplied by 25. To solve for \(m\), you need to perform the inverse operation of multiplication, which is division. By dividing both sides by 25, we get:

• \(m = \frac{225}{25}\)
• Simplified, \(m = 9\)

It's important to divide both sides of the equation by the same number to maintain equality. This step of division not only helps isolate the variable but also simplifies computation, paving the way to finding a correct and valid solution. Always remember: whatever operation you do to one side of the equation, you must do it to the other side as well. This keeps the equation balanced and accurate.

Checking Solutions

After solving an equation, it’s crucial to verify your results by checking the solution. This step ensures that there was no mistake during the solving process.

To check a solution, substitute the calculated value back into the original equation and see if both sides of the equation equal each other. For example, we found that \(m = 9\). Substitute \(m\) back into the original equation \(25m = 225\):

• Substitute: \(25 \times 9 = 225\)
• Result: \(225 = 225\)

The equation holds true, confirming that \(m = 9\) is indeed the correct solution. This step is easy to skip, but it’s vital for confirming accuracy and reinforcing your understanding of equation solving. It assures you that your operations were performed correctly and guards against small mistakes that might have occurred during the initial solution steps.