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Chapter 8: Problem 176

Solve the linear equation using the general strategy. $$0.2(30 n+50)=28$$

Short Answer

Expert verified
n = 3

Step by step solution

01

- Distribute the Constant

Distribute the constant 0.2 to both terms inside the parentheses. This means multiplying 0.2 by each term: \[0.2(30n + 50) = 28\]\[0.2 \times 30n + 0.2 \times 50 = 28\]\[6n + 10 = 28\]
02

- Isolate the Variable Term

Subtract 10 from both sides of the equation to isolate the term with the variable on one side:\[6n + 10 - 10 = 28 - 10\]\[6n = 18\]
03

- Solve for the Variable

Divide both sides of the equation by 6 to solve for the variable n:\[\frac{6n}{6} = \frac{18}{6}\]\[n = 3\]

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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 is a core concept in algebra. It allows us to simplify expressions by distributing a factor over terms inside parentheses. For example, in the equation \(0.2(30n + 50) = 28\), we use the distributive property to multiply 0.2 by each term inside the parentheses.
This gives us:
\[0.2 \times 30n + 0.2 \times 50 = 28\]
After performing these multiplications, the equation simplifies to:
\[6n + 10 = 28\]
By understanding the distributive property, you can handle more complex equations confidently.
Isolate Variable
Once we have applied the distributive property, our next goal is to isolate the variable. This means we want the term with the variable (in this case, \(6n\)) to be by itself on one side of the equation. We do this by performing operations that will eliminate other terms from that side. Here鈥檚 how we do it step by step:
We start with:
\[6n + 10 = 28\]
To isolate \(6n\), subtract 10 from both sides of the equation:
\[6n + 10 - 10 = 28 - 10\]
This simplifies to:
\[6n = 18\]
Now, the variable term \(6n\) is isolated, making it easier to solve for \(n\).
Solve for Variable
The final step in solving a linear equation is to solve for the variable. When we have an equation like \(6n = 18\), we need \(n\) by itself. This is done by dividing both sides of the equation by the coefficient (the number in front of the variable), which is 6 in this case:
\[\frac{6n}{6} = \frac{18}{6}\]
This simplifies to:
\[n = 3\]
So, the variable \(n\) has a value of 3. By following these steps鈥攁pplying the distributive property, isolating the variable, and solving for the variable鈥攜ou can solve most linear equations efficiently.

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