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Chapter 8: Problem 34

In the following exercises, solve each equation. $$ 3(5 n-1)-14 n+9=1-2 $$

Short Answer

Expert verified
n = -7

Step by step solution

01

Simplify the Equation

First, distribute the 3 into the terms inside the parentheses: \[3(5n - 1) = 15n - 3\]So, the equation becomes: \[15n - 3 - 14n + 9 = 1 - 2\]
02

Combine Like Terms

Now, combine the like terms on both sides of the equation:\[15n - 14n - 3 + 9 = -1\]This simplifies to:\[n + 6 = -1\]
03

Solve for the Variable

Isolate the variable by subtracting 6 from both sides of the equation:\[n + 6 - 6 = -1 - 6\]So, we get:\[n = -7\]

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Key Concepts

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

simplifying equations
When faced with a complex equation, the first step is to simplify it. In our exercise, we start with the equation: 3(5n - 1) - 14n + 9 = 1 - 2.Here, we need to simplify the left-hand side. Simplifying means making the equation easier to work with by performing basic operations like distributing and combining like terms. In our example, we distribute the 3 into the terms inside the parentheses: 3(5n - 1) = 15n - 3.Now, replace this back into the equation: 15n - 3 - 14n + 9 = 1 - 2. By performing these operations correctly, we make the problem more manageable. Now, let's move on to combining like terms.
combining like terms
After simplifying the equation, the next step is to combine like terms. Like terms are terms in the equation that have the same variable to the same power. For example, in our equation, 15n - 3 - 14n + 9 = -1, we can identify like terms: - 15n and -14n, because they both have the variable n. - -3 and +9, because they are both constants (numbers without variables). First, combine the like terms with the variable n: 15n - 14n = n. Then, combine the constant terms: -3 + 9 = 6. So now our equation becomes: n + 6 = -1. This makes the equation much simpler to solve. The final step is to isolate the variable.
isolating variables
The last step in solving our equation is to isolate the variable. This means getting the variable (in this case, n) on its own on one side of the equation. Our simplified equation is: n + 6 = -1. To isolate n, we need to get rid of the +6. We can do this by subtracting 6 from both sides of the equation: n + 6 - 6 = -1 - 6. On the left-hand side, the +6 and the -6 cancel each other out, leaving us with: n = -7. And there we go! We've isolated the variable and found that n = -7. Isolating the variable is crucial because it gives us the solution to the equation. Remember, accuracy in each step is key to solving linear equations effectively.

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