Which Of The Following Equations Have Exactly One Solution?Choose All Answers That Apply:A. \[$-6x - 6 = 103x - 103\$\]B. \[$103x - 6 = 103x - 103\$\]C. \[$-103x - 6 = -6x - 103\$\]D. \[$-6x - 6 = -6x - 103\$\]

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Introduction

In mathematics, an equation is a statement that expresses the equality of two mathematical expressions. Equations can have one or more solutions, and in this article, we will explore which of the given equations have exactly one solution. We will analyze each equation and determine whether it has a unique solution or not.

Understanding Linear Equations

Before we dive into the analysis of the given equations, let's briefly review the concept of linear equations. A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, a linear equation is an equation that can be written in the form:

ax + b = c

where a, b, and c are constants, and x is the variable.

Analyzing Equation A

Let's start by analyzing Equation A:

-6x - 6 = 103x - 103

To determine whether this equation has exactly one solution, we need to isolate the variable x. We can do this by adding 6x to both sides of the equation and then subtracting 103x from both sides:

-6x - 6x - 6 = 103x - 103x - 103

This simplifies to:

-12x - 6 = -103

Next, we can add 6 to both sides of the equation to get:

-12x = -97

Now, we can divide both sides of the equation by -12 to solve for x:

x = 97/12

This equation has a unique solution, which is x = 97/12.

Analyzing Equation B

Now, let's analyze Equation B:

103x - 6 = 103x - 103

At first glance, this equation may seem to have a unique solution. However, if we subtract 103x from both sides of the equation, we get:

-6 = -103

This equation is not true, as -6 is not equal to -103. Therefore, Equation B has no solution.

Analyzing Equation C

Next, let's analyze Equation C:

-103x - 6 = -6x - 103

We can start by adding 6x to both sides of the equation to get:

-103x + 6x - 6 = -6x + 6x - 103

This simplifies to:

-97x - 6 = -103

Next, we can add 6 to both sides of the equation to get:

-97x = -97

Now, we can divide both sides of the equation by -97 to solve for x:

x = 97/97

This equation has a unique solution, which is x = 1.

Analyzing Equation D

Finally, let's analyze Equation D:

-6x - 6 = -6x - 103

If we add 6x to both sides of the equation, we get:

-6 - 6 = -6x - 6x - 103

This simplifies to:

-12 = -12x - 103

Next, we can add 103 to both sides of the equation to get:

-9 = -12x

Now, we can divide both sides of the equation by -12 to solve for x:

x = 9/12

This equation has a unique solution, which is x = 3/4.

Conclusion

In conclusion, the equations that have exactly one solution are:

  • Equation A: -6x - 6 = 103x - 103
  • Equation C: -103x - 6 = -6x - 103
  • Equation D: -6x - 6 = -6x - 103

These equations have unique solutions, which are x = 97/12, x = 1, and x = 3/4, respectively. On the other hand, Equation B has no solution.

Final Thoughts

In this article, we analyzed four different equations and determined which ones have exactly one solution. We used algebraic techniques to isolate the variable x and solve for its value. By following these steps, we can determine whether an equation has a unique solution or not. This is an important concept in mathematics, as it helps us understand the behavior of equations and make predictions about their solutions.

Introduction

In our previous article, we explored which of the given equations have exactly one solution. We analyzed each equation and determined whether it has a unique solution or not. In this article, we will answer some frequently asked questions about equations with exactly one solution.

Q: What is an equation with exactly one solution?

A: An equation with exactly one solution is an equation that has a unique value for the variable. In other words, there is only one possible value for the variable that satisfies the equation.

Q: How do I determine if an equation has exactly one solution?

A: To determine if an equation has exactly one solution, you need to isolate the variable and solve for its value. If the equation has a unique solution, it means that there is only one possible value for the variable that satisfies the equation.

Q: What are some common characteristics of equations with exactly one solution?

A: Equations with exactly one solution often have the following characteristics:

  • They have a unique solution, which means that there is only one possible value for the variable that satisfies the equation.
  • They are linear equations, which means that the highest power of the variable is 1.
  • They have a constant term on both sides of the equation, which means that the equation is in the form ax + b = c.

Q: Can an equation have more than one solution?

A: Yes, an equation can have more than one solution. This is known as a system of equations, where there are multiple equations with multiple variables. In this case, the equations may have multiple solutions, or they may have no solution at all.

Q: Can an equation have no solution?

A: Yes, an equation can have no solution. This is known as a contradiction, where the equation is false and has no possible solution.

Q: How do I know if an equation has no solution?

A: To determine if an equation has no solution, you need to check if the equation is a contradiction. If the equation is a contradiction, it means that there is no possible solution for the variable.

Q: What are some common mistakes to avoid when solving equations with exactly one solution?

A: Some common mistakes to avoid when solving equations with exactly one solution include:

  • Not isolating the variable correctly
  • Not checking for contradictions
  • Not considering the possibility of multiple solutions

Q: How do I apply what I've learned to real-world problems?

A: To apply what you've learned to real-world problems, you need to be able to analyze and solve equations with exactly one solution. This involves using algebraic techniques to isolate the variable and solve for its value. By following these steps, you can determine whether an equation has a unique solution or not.

Conclusion

In conclusion, equations with exactly one solution are an important concept in mathematics. By understanding how to determine if an equation has a unique solution, you can apply what you've learned to real-world problems. Remember to avoid common mistakes, such as not isolating the variable correctly or not checking for contradictions. With practice and patience, you can become proficient in solving equations with exactly one solution.

Final Thoughts

In this article, we answered some frequently asked questions about equations with exactly one solution. We covered topics such as what an equation with exactly one solution is, how to determine if an equation has exactly one solution, and common characteristics of equations with exactly one solution. By following these steps and avoiding common mistakes, you can become proficient in solving equations with exactly one solution.