Problem 41 - Prealgebra

Question

Solve each equation by finding a number to replace \(n\) that will make the equation a true statement. $$650=10 \cdot n$$

Step-by-Step Solution

Answer

The value of \(n\) is 65.

Step 1: Understand the equation

The given equation is \(650 = 10 \cdot n\). Our goal is to find the value of \(n\) that will satisfy this equation.

Step 2: Isolate the variable

To find \(n\), we need to get \(n\) by itself on one side of the equation. Since \(n\) is currently multiplied by 10, we will divide both sides of the equation by 10 to isolate \(n\).

Step 3: Perform the division

Divide both sides by 10:\[ \frac{650}{10} = \frac{10 \cdot n}{10} \]This simplifies to:\[ 65 = n \]

Step 4: Verify the solution

Substitute \(n = 65\) back into the original equation to verify:\(10 \cdot 65 = 650\). This simplifies to 650 = 650, confirming our solution is correct.

Key Concepts

Isolate the Variable

To solve linear equations, the primary goal is usually to isolate the variable. This means you want to get the variable all by itself on one side of the equation. In the equation \(650 = 10 \cdot n\), our variable is \(n\). To isolate \(n\), you need to undo any operations that are being performed on it. In this case, \(n\) is being multiplied by 10.

• Identify the operation affecting the variable. Here, it's multiplication by 10.
• To undo this operation, use the opposite operation—in this case, division.
• This will allow you to solve for \(n\) by itself.

Once these steps are performed, you have successfully isolated the variable, which is a crucial step in solving linear equations.

Verify the Solution

Verifying the solution of an equation helps ensure that the value found for the variable makes the entire equation true. Once you've isolated the variable and found \(n = 65\) in our example, the next step is to verify the solution.

• Substitute the value back into the original equation: \(10 \cdot 65 = 650\).
• Calculate the expression on both sides: \(650 = 650\).
• If both sides of the equation are equal, then your solution is correct.
• If they are not equal, re-evaluate the steps taken to find the error.

Verification is vital because it confirms that the solution fits the original problem requirements and adheres to mathematical laws.

Division in Equations

Division is a common and essential operation when solving equations, especially for isolating variables. In the given equation \(650 = 10 \cdot n\), we needed to perform division to solve for the variable \(n\). Here’s a breakdown of why and how division is used:

• Identify the multiplication: \(n\) is being multiplied by 10, so you will divide by 10 to isolate \(n\).
• Divide each side by the same non-zero number to maintain equality: \( \frac{650}{10} = \frac{10 \cdot n}{10} \).
• Simplify both sides to find \(n = 65\).
• Always ensure to divide by a non-zero number to keep the equation valid.

Through division, not only do you isolate the variable but also maintain the equivalence of the equation, which is fundamental in solving linear equations successfully.