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Chapter 8: Problem 168

Solve the linear equation using the general strategy. $$-4(2 n+1)=16$$

Short Answer

Expert verified
n = -\frac{5}{2}

Step by step solution

01

Distribute the multiplication

Multiply \( -4 \) by both terms inside the parenthesis \( 2n + 1 \): \( -4 \times 2n \) and \( -4 \times 1 \). This gives \[-4(2n + 1) = -8n - 4.\]
02

Set up the equation

Rewrite the equation with the distributed values: \[-8n - 4 = 16.\]
03

Isolate the variable term

Add 4 to both sides of the equation to isolate the variable term: \[-8n - 4 + 4 = 16 + 4,\] simplifying to \[-8n = 20.\]
04

Solve for the variable

Divide both sides by \( -8 \) to solve for \( n \): \[\frac{-8n}{-8} = \frac{20}{-8},\] which simplifies to \[n = -\frac{20}{8},\] and reducing the fraction we get \[n = -\frac{5}{2}.\]

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Key Concepts

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

distributive property
The distributive property helps to simplify expressions and solve equations. When you see an expression like \(-4(2n + 1)\), you apply multiplication to each term inside the parentheses.
For instance, \(-4 \times 2n\) turns into \(-8n\), and \(-4 \times 1\) becomes \(-4\).
This lets you rewrite the original expression without parentheses:
\[-4(2n + 1) = -8n - 4\]
Recognizing and applying the distributive property correctly is essential in prealgebra. It will simplify the process of solving equations, especially linear ones.
isolating variable
Isolating the variable means getting the unknown by itself on one side of the equation. This often needs you to perform operations on both sides equally.
Using our example \(-8n - 4 = 16\), you want to isolate \(n\):
  • Add 4 to both sides: \(-8n - 4 + 4 = 16 + 4\)
  • Your equation now looks like: \[-8n = 20\]
  • Divide both sides by \(-8\): \(-8n / -8 = 20 / -8\)
  • Now you have: \[-n = -2.5\]

Isolating the variable is a critical step toward finding your solution. It often involves simple arithmetic but requires careful steps to ensure both sides remain balanced.
simplifying fractions
Simplifying fractions makes your final answer easier to understand. Once you isolate the variable, you might end up with a fraction to simplify.
From \(n = -20/8\), you can reduce it by finding the greatest common divisor.
\[-20/8 = -5/2\]
Here, both 20 and 8 are divisible by 4.
Always check if you can simplify fractions for a cleaner answer. This small step is often crucial in prealgebra problems.
prealgebra
Prealgebra lays the foundation for all future algebra concepts. Its focus is on understanding variables, simple equations, fractions, and basic arithmetic properties.
Let's recap using our example equation \(-4(2n + 1) = 16\):
  • Use the distributive property: \(-8n - 4 = 16\)
  • Isolate the variable: \(-8n = 20\)
  • Simplify the fraction: \(n = -5/2\)
Prealgebra equips you with the tools to solve much more complex equations in the future. Mastery of these fundamental steps will open doors to understanding algebra, and beyond.

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Most popular questions from this chapter

In the following exercises, solve each equation by clearing the fractions. $$ \frac{2}{5} n-\frac{1}{10}=\frac{7}{10} $$

In the following exercises, solve each equation by clearing the decimals. $$ 0.6 p-1.9=0.78 p+1.7 $$

Is \(-18\) a solution to the equation \(3 x=16-5 x ?\) How do you know?

In the following exercises, translate to an equation and then solve. The sum of \(-4 n\) and 5\(n\) is \(-32\)

In the following exercises, solve each equation using the Division Property of Equality. $$ -y=4 $$
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