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Chapter 2: Problem 63

Solve each equation, and check your solution. $$ 9(v+1)-3 v=2(3 v+1)-8 $$

Short Answer

Expert verified
There is no solution, as the equation simplifies to a false statement.

Step by step solution

01

- Distribute

Apply the distributive property to both sides of the equation. Distribute the constants across the parentheses: \[ 9(v + 1) - 3v = 2(3v + 1) - 8 \]This simplifies to: \[ 9v + 9 - 3v = 6v + 2 - 8 \]
02

- Combine Like Terms

Combine the like terms on both sides to simplify the equation: \[ (9v - 3v) + 9 = 6v + (2 - 8) \]This simplifies to: \[ 6v + 9 = 6v - 6 \]
03

- Isolate the Variable

Move all terms involving the variable to one side of the equation. Subtract 6v from both sides: \[ 6v + 9 - 6v = 6v - 6 - 6v \]This simplifies to: \[ 9 = -6 \]
04

- Analyze the Result

Since the equation simplifies to \[ 9 = -6 \], which is a false statement, there is no solution. This means the original equation has no solutions.
05

- Check the Solution

Since there is no solution, there is nothing to check. The equation is inconsistent.

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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 useful algebraic rule for simplifying expressions. It says that you can multiply a number by a group of numbers in parentheses: a(b + c) = ab + acIn the exercise given, we began by distributing the constants across the parentheses on both sides of the equation:9(v + 1) - 3v = 2(3v + 1) - 8Applying the distributive property, you get:9v + 9 - 3v = 6v + 2 - 8This simplifies the equation, making it easier to solve.
Combining Like Terms
After using the distributive property, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power. Here, we combine the like terms on each side of the equation:(9v - 3v) + 9 = 6v + (2 - 8)Simplifying these, we get:6v + 9 = 6v - 6By combining the terms that contain the variable 'v' and the constant numbers, the equation is further simplified.
Isolating the Variable
To solve for the variable, we need to isolate it on one side of the equation. This makes it easier to see what the value would be. In our example, we subtract 6v from both sides to try and isolate 'v':6v + 9 - 6v = 6v - 6 - 6vThis simplifies to:9 = -6However, this doesn't allow us to isolate the variable, and instead leads us to the conclusion that something is inconsistent in our equation.
Inconsistent Equations
An inconsistent equation is one that has no solutions. This happens when you end up with a statement that is false, such as 9 = -6. In the context of our exercise, after isolating the variables, we ended up with a false statement indicating that no value of 'v' can satisfy the original equation. Therefore, the given equation is inconsistent and has no solution.

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