Solving Equations: How Many Solutions Exist?

Hey guys! Let's dive into the world of equations and figure out how many solutions a particular equation can have. Today, we're tackling the equation: -12z - 1 - z = -12z - 20. Our mission is to determine if this equation has no solution, one solution, or infinitely many solutions. This is a fundamental concept in algebra, and understanding it will help you tackle more complex problems down the road. So, grab your pencils, and let’s get started!

Understanding Solutions to Equations

Before we jump into solving the equation, let's quickly recap what it means for an equation to have different types of solutions. When we talk about the solutions of an equation, we're referring to the values of the variable (in this case, 'z') that make the equation true. There are three possible scenarios:

• No Solution: This means there is no value for the variable that will satisfy the equation. The equation will always be false, no matter what value you plug in for 'z'.
• One Solution: This means there is exactly one value for the variable that makes the equation true. This is the most common scenario you'll encounter.
• Infinitely Many Solutions: This means that any value you substitute for the variable will make the equation true. The left side of the equation is essentially equivalent to the right side.

Knowing these definitions is crucial because it gives us a framework for interpreting the results we obtain when solving the equation. Now that we're all on the same page, let’s proceed with solving the given equation step by step.

Step-by-Step Solution

Alright, let’s break down the equation -12z - 1 - z = -12z - 20 step by step to determine the number of solutions. Our goal is to simplify the equation and isolate the variable 'z' on one side. This will help us see clearly what values of 'z', if any, satisfy the equation. Here's how we can do it:

1. Combine Like Terms

The first step is to combine like terms on each side of the equation. On the left side, we have two terms with 'z': -12z and -z. Combining these, we get:

-12z - z = -13z

So, the left side of the equation simplifies to -13z - 1. The right side of the equation, -12z - 20, already has its terms simplified.

2. Rewrite the Equation

Now, let's rewrite the equation with the simplified left side:

-13z - 1 = -12z - 20

This looks a bit cleaner, doesn't it? We're one step closer to figuring out the solution.

3. Isolate the Variable Term

Next, we want to get all the terms with 'z' on one side of the equation. To do this, we can add 12z to both sides. This will eliminate the -12z term on the right side:

-13z - 1 + 12z = -12z - 20 + 12z

Simplifying both sides, we get:

-z - 1 = -20

We've successfully moved the 'z' terms to the left side. Now, let’s isolate 'z' completely.

4. Isolate the Constant Term

To further isolate 'z', we need to get rid of the constant term (-1) on the left side. We can do this by adding 1 to both sides of the equation:

-z - 1 + 1 = -20 + 1

Simplifying, we have:

-z = -19

Almost there! We just need to get rid of that negative sign on the 'z'.

5. Solve for z

To solve for 'z', we can multiply both sides of the equation by -1. This will change the sign of both terms:

(-1) * (-z) = (-1) * (-19)

This gives us:

z = 19

And there we have it! We've successfully solved for 'z'. Now, let’s interpret our result and see what it means for the number of solutions.

Determining the Number of Solutions

So, after all that simplification and isolation, we've arrived at the solution: z = 19. What does this tell us about the number of solutions the original equation has? Well, we've found one specific value for 'z' that makes the equation true. This means that there is one solution to the equation.

To double-check our work, we can substitute z = 19 back into the original equation and see if it holds true:

-12(19) - 1 - (19) = -12(19) - 20

Let's simplify:

-228 - 1 - 19 = -228 - 20

-248 = -248

Yep, it checks out! The equation is true when z = 19. This confirms that our solution is correct and that the equation has exactly one solution.

Why Only One Solution?

You might be wondering why this equation has only one solution, while others might have none or infinitely many. The key is in how the equation simplifies. In this case, when we isolated the variable 'z', we ended up with a unique value for 'z'. There wasn't any contradiction (which would mean no solution) or any identity (which would mean infinitely many solutions). We landed on a single, specific value, indicating a single solution.

Conclusion

In summary, the equation -12z - 1 - z = -12z - 20 has one solution, and that solution is z = 19. We arrived at this conclusion by systematically simplifying the equation, isolating the variable, and solving for its value. We also verified our solution by plugging it back into the original equation. Understanding how to solve equations and determine the number of solutions is a crucial skill in algebra, and I hope this step-by-step explanation has helped you grasp the concept a little better. Keep practicing, guys, and you'll become equation-solving pros in no time!