Problem 42 - Prealgebra

Question

Solve each equation by finding a number to replace \(n\) that will make the equation a true statement. $$630=7 \cdot n$$

Step-by-Step Solution

Answer

The value of \( n \) that makes the equation true is 90.

Step 1: Understand the Equation

The given equation is \( 630 = 7 \cdot n \). We need to find the value of \( n \) such that the equation holds true.

Step 2: Isolate the variable

To find \( n \), we need to isolate it on one side of the equation. We do this by dividing both sides of the equation by 7, which is the coefficient of \( n \).

Step 3: Perform Division

Divide 630 by 7: \[ n = \frac{630}{7} \]. Calculating this gives \( n = 90 \).

Step 4: Verify the Solution

Substitute \( n = 90 \) back into the original equation to check if the equation is correct: \( 630 = 7 \cdot 90 \). Simplifying the right side gives \( 630 = 630 \), which is a true statement.

Key Concepts

Isolation of Variable

In equation solving, isolating the variable is a fundamental step. It means transforming the equation so the unknown variable is by itself on one side of the equation. This helps you easily determine its value.

- Start by identifying the term with the variable.- In our original problem, the term is \(7 \cdot n\).- To isolate \(n\), perform the opposite operation of multiplication, which is division.

Isolation requires reversing operations:- For addition, use subtraction.- For multiplication, use division, like we did here.This simplifies the problem and paves the way for finding the solution easily.

Division in Equations

Division is one of the key operations in solving equations, especially those involving a variable.When isolating a variable that is being multiplied by a coefficient, division is employed to 'break' this multiplication.

In the problem, we divided both sides of the equation by 7, the coefficient of \(n\). This step looks like:- \[ n = \frac{630}{7} \]Performing the division yields \(n = 90\). This simplification step ensures clarity and precision when solving for unknowns.

Always check if the division is possible:- If the dividend (like 630 here) is not perfectly divisible by the divisor (7 in this case), consider simplifying or reassessing your methods. This ensures accuracy in your calculations and conclusions.

Verifying Solutions

After finding a solution to an equation, it's crucial to verify it. This ensures that the solution indeed satisfies the original equation, confirming its correctness.

Let's see how we do this:- Substitute the obtained value of \(n = 90\) back into the original equation.- Check the equation: \(630 = 7 \cdot 90\).

Simplify the right-hand side:- \(7 \cdot 90 = 630\)- Thus, \(630 = 630\), confirming our solution is correct.

Verifying:- Helps avoid errors.- Provides confidence in your solution.- Encourages a deeper understanding of equation manipulation.This step reinforces the logical consistency of your answer.