Problem 53 - Prealgebra

Question

Solve each equation. Check your solution. $$29=n+4$$

Step-by-Step Solution

Answer

The solution is \( n = 25 \).

Step 1: Rearrange the Equation

To isolate the variable, you need to get \( n \) by itself on one side of the equation. The equation given is: \( 29 = n + 4 \). Subtract 4 from both sides of the equation to do this: \( 29 - 4 = n \).

Step 2: Solve for the Variable

Perform the subtraction on the left side: \( 29 - 4 = 25 \). Therefore, \( n = 25 \). This is the value of \( n \) that satisfies the equation.

Step 3: Check Your Solution

To ensure that \( n = 25 \) is correct, substitute it back into the original equation. Replace \( n \) with 25: \( 29 = 25 + 4 \). Now compute \( 25 + 4 \) to check that it equals 29: \( 25 + 4 = 29 \), which confirms the solution is correct.

Key Concepts

Variable Isolation

In pre-algebra, solving an equation usually starts with isolating the variable. This means you want to have the variable by itself on one side of the equation. For the equation given, \( 29 = n + 4 \), isolating \( n \) requires eliminating the constant 4 that is on the same side of the equation as \( n \). Here's how you can do it:

• Identify the constant term on the side with the variable, which in this case is 4.
• Decide how to "get rid of" this constant. Since 4 is added to \( n \), you'll do the opposite operation (subtract 4) from both sides of the equation.

This results in the operation: \( 29 - 4 = n \). Now you have successfully isolated the variable \( n \), as it stands alone on one side of the equation. This step is crucial because it sets the stage for finding the variable's actual value.

Checking Solutions

After you find a value for the variable, it's important to confirm that it's correct. This is done by checking the solution. Checking solutions ensures that the operations you performed led to a valid answer. To check:

• Substitute the calculated value back into the original equation. For our problem, substitute \( n = 25 \) back into the equation \( 29 = n + 4 \).
• Replace \( n \) with 25, giving: \( 29 = 25 + 4 \).
• Calculate \( 25 + 4 \) to see if it equals the other side of the equation, which is 29.

Since \( 25 + 4 = 29 \) is true, this confirms that our solution \( n = 25 \) is correct. This step provides assurance that the isolation of the variable and arithmetic were done accurately.

Arithmetic Operations

Arithmetic operations are the foundation of solving equations. In this problem, the arithmetic operation used is subtraction. Here's how it played out:

• To isolate \( n \), you need to subtract 4 from both sides of the equation \( 29 = n + 4 \). This gives you \( 29 - 4 = n \).
• Perform the subtraction, \( 29 - 4 = 25 \), and the result gives you the value of \( n \).

Subtraction here is used to "undo" the addition of 4 to \( n \). Understanding the inverse relationship between adding and subtracting is what allows equations to be manipulated and solved. With arithmetic, you can systematically unravel equations to find the values of unknowns.