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Chapter 2: Problem 94

Solve. $$ 2 n-6=12 n+1 $$

Short Answer

Expert verified
The solution is \( n = -\frac{7}{10} \).

Step by step solution

01

Get the Variables on One Side

First, let's move the variable terms to one side of the equation. Subtract \(12n\) from both sides:\[2n - 12n - 6 = 12n - 12n + 1\]This simplifies to:\[-10n - 6 = 1\]
02

Isolate the Variable Term

Add 6 to both sides to move the constant term to the right side of the equation:\[-10n - 6 + 6 = 1 + 6\]This simplifies to:\[-10n = 7\]
03

Solve for the Variable

Divide both sides by \(-10\) to solve for \(n\):\[\frac{-10n}{-10} = \frac{7}{-10}\]This simplifies to:\[n = -\frac{7}{10}\]

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Key Concepts

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

Understanding Algebra
Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It can simplify complex problems and solve for unknown values, represented by variables like \( n \) in our exercise.
To solve algebraic equations often involves a series of steps:
  • Identifying the equation to solve.
  • Performing operations to both sides of the equation.
  • Simplifying expressions systematically.
Algebra is all about balance. You perform the same operations on both sides of the equation to maintain equality. This is key when manipulating expressions to isolate the variable.
Variable Isolation Technique
Variable isolation involves rearranging an equation so the variable of interest stands alone on one side of the equation. This is essential to finding its actual value.
Let's break it down:
  • Identify which term contains the variable you need to isolate.
  • Move all terms not containing the variable to the other side. This often involves adding, subtracting, multiplying, or dividing both sides of the equation.
In our exercise, we moved terms around to get \(-10n = 7\). Here, \(n\) is isolated except for the negative coefficient. The next step involves solving for \(n\).
Practicing variable isolation will strengthen your algebra skills and allow you to effortlessly tackle more complex problems.
Equation Simplification Process
Simplification in algebra means transforming an equation into an equivalent form that is easier to work with. This usually involves combining like terms and making calculations easier.For simplification, consider these steps:
  • Identify and group like terms on the same side of the equation.
  • Perform arithmetic operations to combine terms and simplify the expression.
In our problem, after bringing variables to one side, we simplified \(2n - 12n - 6 = 1\) to \(-10n - 6 = 1\). Further operations led to \(n = -\frac{7}{10}\), providing a cleaner, straightforward solution.
Mastering equation simplification helps avoid errors and makes solving equations much more manageable.

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Most popular questions from this chapter

Graph all solutions on a number line and provide the corresponding interval notation. $$ x<-1 $$

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Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 3 x<18 \text { and } 5 x>-20 $$

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -2 x+1<-1 \quad \text { or }-2 x+1>1 $$
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