Problem 23 - Elementary Algebra

Question

Solve. $$ 53 n+3=23 $$

Step-by-Step Solution

Answer

n = \(\frac{20}{53}\)

Step 1: Isolate the Variable Term

Start with the given equation:\[ 53n + 3 = 23 \]Your goal is to solve for \( n \). The first step is to isolate the term with \( n \) by subtracting 3 from both sides:\[53n + 3 - 3 = 23 - 3\]This simplifies to:\[53n = 20\]

Step 2: Solve for the Variable

Now divide both sides by 53 to get \( n \) by itself:\[\frac{53n}{53} = \frac{20}{53}\]This simplifies to:\[n = \frac{20}{53}\]

Key Concepts

Isolate the Variable

When it comes to solving linear equations, one of the key techniques is to isolate the variable. This simply means getting the variable (in this case, \( n \)) by itself on one side of the equation. For our example, the equation starts as \( 53n + 3 = 23 \). To isolate \( n \), you need to remove any numbers or coefficients that are on the same side of the equation as the variable.

Here’s how it works:

• Look at what’s being added or subtracted from the variable. In our equation, \( 3 \) is added to \( 53n \).
• To get rid of \( +3 \), do the inverse operation. Subtract \( 3 \) from both sides to maintain balance. This gives you \( 53n = 20 \).
• Now, \( n \) is nicely by itself along with its coefficient, making it easier to solve for \( n \).

By mastering how to isolate the variable, you lay the groundwork for solving any linear equation effectively. This step is crucial because it simplifies the equation and sets you up for the next phase: solving for the variable.

Solving Equations

After isolating the variable, the next big step is solving the equation to find the exact value of the variable. Once you have simplified the equation to \( 53n = 20 \), the next question is: what does \( n \) equal?

To find \( n \), you do the following:

• You need to "undo" the multiplication of \( n \) by 53. The opposite of multiplication is division.
• Divide both sides of the equation by 53 to isolate \( n \). This results in \( n = \frac{20}{53} \).
• Now, \( n \) has been solved, and its value is \( \frac{20}{53} \).

This simple division is the final step of solving the equation once the variable is isolated. Remember that the equation should remain balanced, which means whatever you do to one side, you must do to the other. Solving equations might seem daunting at first, but with practice and by following these logical steps, it becomes much easier.

Algebraic Manipulation

Algebraic manipulation is the set of techniques used to move and change terms within an equation to simplify it or solve for variables. It's a fundamental skill in algebra and vital for solving almost any equation or formula. Let's break down this concept with the given example: \( 53n + 3 = 23 \).

Key operations of algebraic manipulation include:

• Adding and Subtracting: To change the equation, such as subtracting 3 (i.e., \( 53n + 3 - 3 = 23 - 3 \)), which simplifies to \( 53n = 20 \). The aim is to reduce and simplify the equation step-by-step.
• Multiplying and Dividing: These are used to eliminate coefficients. For \( n \), divide both sides by 53 to find \( n = \frac{20}{53} \). Multiplication or division can simplify equations further.
• Balanced Equations: Ensure whatever operation is used on one side is equally applied to the other side to maintain equality.

Mastering algebraic manipulation paves the way for solving more complex problems with confidence. Each successful manipulation of the original equation gets you closer to the solution while maintaining the balance that the equation dictates.