Problem 21 - Elementary Algebra

Question

Solve. $$ -40 n=-53 $$

Step-by-Step Solution

Answer

The solution is \( n = \frac{53}{40} \).

Step 1: Identify the Equation

The given equation is \[-40n = -53\] This is a linear equation where the goal is to solve for the variable \( n \).

Step 2: Isolate the Variable

To solve for \( n \), we need to isolate it on one side of the equation. We will do this by dividing both sides of the equation by \(-40\):\[\frac{-40n}{-40} = \frac{-53}{-40}\]

Step 3: Simplify the Equation

Simplify the left side of the equation by canceling out \(-40\) with itself, which leaves us with \( n \). Then simplify the right side of the equation:\[n = \frac{53}{40}\]

Step 4: Finalize the Answer

The simplified form \( \frac{53}{40} \) can remain as a fraction because it's not easily reducible and is already simplified. Thus, this is our solution for \( n \).

Key Concepts

Solving Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed. They are in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Solving these equations involves finding the value of \( x \) that makes the equation true. This is akin to uncovering the 'mystery' value that balances both sides of the equation.

To solve a linear equation, follow these general steps:

• Identify the equation structure to understand which parts involve variables and constants.
• Use operations like addition, subtraction, multiplication, and division to simplify the equation.
• Perform each operation on both sides to maintain the equation's balance.

Once these steps are applied, the variable should be isolated, and you can solve for its value, just like we solved for \( n \) in the provided exercise.

Isolation of Variables

Isolation of variables is a crucial step in solving linear equations. The aim here is to get the variable alone on one side of the equation, free from coefficients or other numbers.

In practice, this involves:

• Identifying the coefficient attached to the variable. In our example, \(-40\) was the coefficient of \( n \).
• Performing inverse operations to cancel out coefficients or constants that are on the same side as the variable. Inverse operations are ones that do the opposite: addition vs. subtraction, and multiplication vs. division.
• Applying the inverse operation equally to both sides of the equation.

For the exercise, we divided both sides by \(-40\) to isolate \( n \), effectively removing the coefficient. This step is essential because it simplifies our focus to solving for just the variable, clearing away obstacles in the equation.

Simplifying Fractions

Simplifying fractions is often the final touch needed to arrive at a clean, intelligible solution when dealing with linear equations. Simplifying refers to reducing a fraction to its smallest form where the top and bottom numbers have no common divisors other than 1.

To simplify fractions:

• Check the numerator and denominator for any common divisors.
• If there are common divisors, divide both the numerator and the denominator by this number.
• If no simplification is possible (i.e., the fraction is already in its simplest form), the fraction stands as is.

In our exercise, the fraction \( \frac{53}{40} \) couldn't be simplified further, as 53 and 40 share no divisors besides 1. Therefore, it served as our final solution for \( n \). Simplifying fractions ensures clarity and precision in mathematical solutions, allowing others to easily interpret and verify the results.