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Chapter 2: Problem 17

Solve each linear equation. $$ 3(4 n-1)-2=8 n+3 $$

Short Answer

Expert verified
n = 2

Step by step solution

01

- Distribute the 3

Apply the distributive property to the term on the left-hand side: 3(4n - 1) becomes 12n - 3. So the equation is: 12n - 3 - 2 = 8n + 3.
02

- Combine Like Terms

Combine the constant terms on the left-hand side:12n - 5 = 8n + 3.
03

- Isolate Variable Terms

Subtract 8n from both sides to isolate terms with the variable on one side:4n - 5 = 3.
04

- Isolate the Variable

Add 5 to both sides to isolate the term with the variable:4n = 8.
05

- Solve for n

Divide both sides by 4 to solve for n:n = 2.

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Key Concepts

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

Distributive Property
The first step in solving the given linear equation is to apply the distributive property. This property allows you to simplify expressions like \(a(b + c)\) into \(ab + ac\). In our equation, we have \(3(4n - 1)\).
By applying the distributive property, we multiply 3 by each term inside the parentheses, giving us:
\[3(4n) - 3(1) = 12n - 3\]
Now, the equation looks like this:
\(12n - 3 - 2 = 8n + 3\).
Remember, distributing helps us eliminate the parentheses, simplifying the equation so we can solve it more easily.
Combine Like Terms
Next, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, \(12n\) and \(8n\) are like terms because they both have the variable \(n\). On the other hand, \(-3\) and \(-2\) are like terms because they are both constants.
Combining these terms simplifies the equation further. We update our equation:
\[12n - 3 - 2 = 8n + 3\]
Combine the constant terms on the left-hand side:
\[12n - 5 = 8n + 3\]
This step makes the equation easier to manage by reducing the number of separate terms we need to work with.
Isolate Variable
Now, we need to isolate the variable on one side of the equation to solve for \(n\). We start with our simplified equation:
\[12n - 5 = 8n + 3\]
Subtract \(8n\) from both sides to isolate the variable terms on one side:
\[4n - 5 = 3\]
Next, add 5 to both sides to isolate the term with the variable:
\[4n = 8\]
Finally, divide both sides by 4 to solve for \(n\):
\[n = 2\]
By following these steps, we have successfully isolated the variable and found the solution to the equation. Isolating the variable is key to solving any linear equation because it lets us find the specific value of the unknown.

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