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Chapter 2: Problem 43

Solve each equation. $$3(5 n-1)-14 n+9=10(n-4)-6 n-4(n+1)$$

Short Answer

Expert verified
The solution is \(n = -50\).

Step by step solution

01

Distribute

First, distribute the constants through the parentheses: \(3(5n - 1) - 14n + 9 = 10(n - 4) - 6n - 4(n + 1)\) \(15n - 3 - 14n + 9 = 10n - 40 - 6n - 4n - 4\)
02

Combine Like Terms

Next, combine like terms on both sides of the equation: \((15n - 14n) - 3 + 9 = (10n - 6n - 4n) - 40 - 4\) \(n + 6 = 0 - 44\) \(n + 6 = -44\)
03

Solve for n

Finally, solve for \(n\) by isolating the variable: \(n = -44 - 6\) \(n = -50\)

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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 simplify equations. It says there's no difference between multiplying a number by a group of added numbers and multiplying the number by each added number separately, then adding the results. For example, let's look at our equation:
3(5n - 1): We multiply 3 with both 5n and -1. This gives us 15n - 3. Do the same for each set of parentheses. You get:
3(5n - 1) - 14n + 9 = 10(n - 4) - 6n - 4(n + 1)
15n - 3 - 14n + 9 = 10n - 40 - 6n - 4n - 4
Combining Like Terms
After distributing, combine like terms. Like terms are terms that have the same variable raised to the same power. Let's see it in action:
On the left side, we have: 15n - 14n - 3 + 9. Combine the n's (15n - 14n = n) and the constants (-3 + 9 = 6), yielding n + 6.
On the right side, look at 10n - 40 - 6n - 4n - 4. Combine the n terms (10n - 6n - 4n = 0) and the constants (-40 - 4 = -44). This gives 0 - 44 = -44. Now, your equation looks like:
n + 6 = -44
Isolating Variables
To find the value of the variable, isolate it. This means getting the variable alone on one side of the equation. In our case, let's isolate n:
We have n + 6 = -44. Subtract 6 from both sides to isolate n:
n + 6 - 6 = -44 - 6
n = -50
Linear Equations
Finally, let鈥檚 discuss linear equations. These are equations of the first degree, which means they have variables raised to the power of 1. Our example is a linear equation because n is not squared or cubed.
Solving linear equations involves basic steps:
  • First, distribute any constants.
  • Then, combine like terms.
  • Finally, isolate the variable.
Using these steps, you can solve equations quickly and accurately. So in summary, we found n = -50 by following the steps mentioned. Keep practicing, and you'll master solving linear equations in no time!

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