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Chapter 1: Problem 77

Solve each equation. $$13 n-18=5 n+32$$

Short Answer

Expert verified
The solution is \( n = \frac{25}{4} \) or \( n = 6.25 \).

Step by step solution

01

Combine Like Terms

Start by moving all terms involving the variable \( n \) to one side of the equation. Subtract \( 5n \) from both sides: \[ 13n - 5n - 18 = 5n - 5n + 32 \] which simplifies to: \[ 8n - 18 = 32 \]
02

Isolate the Variable Term

To isolate the term with \( n \), move the constant to the other side by adding 18 to both sides: \[ 8n - 18 + 18 = 32 + 18 \] This gives us: \[ 8n = 50 \]
03

Solve for the Variable

Now, solve for \( n \) by dividing both sides by 8: \[ \frac{8n}{8} = \frac{50}{8} \] So, \( n = \frac{50}{8} \) which simplifies to \( n = \frac{25}{4} \) or \( n = 6.25 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
Linear equations are foundational in algebra, typically formed as expressions equaling each other, where the solutions are values of the variables that make the equation true. The main goal in solving a linear equation, like the one given, is to find the specific value for the variable. This requires balancing the equation: whatever operation you perform on one side, you must do to the other.
  • Start by identifying and grouping terms that are similar, known as 'like terms'.
  • Rearrange the equation to get all variable terms on one side and constants on the other.
  • Perform operations systematically to simplify the equation and get your answer.
Understanding how and when to use these techniques helps in creating a step-by-step path to the solution. You can begin practicing by writing each step clearly, which strengthens your overall math skills.
Isolation of Variables
Isolation of variables is a core step when solving equations. It means getting the variable 鈥� in this case, represented as 'n' 鈥� entirely alone on one side of the equals sign. This often requires moving constants or other variables across the equation:
  • Use addition or subtraction to remove any numbers from the side with your variable.
  • Addition undoes subtraction; thus, you applied addition of 18 to both sides in this exercise.
  • Keep your balance: remember, your goal is to achieve something like 'n = something'.
Once the variable is isolated, you'll be able to clearly see the value that satisfies the equation. It's about breaking down each operation into digestible steps and ensuring each action is mirrored on both sides of the equation.
Combining Like Terms
Combining like terms is simplifying your equation by putting together terms that contain the same variable raised to the same power. This makes the problem easier to handle and helps clarify what needs to be solved.Consider the terms \(13n\) and \(5n\) found in the original equation. Both contain the variable \(n\), so you combine these to simplify:
  • Combine terms by performing the arithmetic operation indicated: in this case, subtraction.
  • This simplifies the equation, reducing it into a more understandable format, like \(8n - 18 = 32\).
  • The resulting simpler equation is much easier to solve or manipulate further.
Successfully combining like terms is often your first major push towards efficiently solving the equation, paving the way for the isolation and final solving steps.

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