Understanding the Concept of One Solution

Oct 13, 2025 by recoveryshake.com 12 views

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. When solving systems of equations, we often encounter different types of solutions, including one solution, no solution, and infinitely many solutions. In this article, we will focus on understanding systems of equations with one solution, specifically the equation $3x + 18 = 5x + 8$ . We will analyze the equation, determine what it means to have one solution, and provide a step-by-step solution to the equation.

Analyzing the Equation

The given equation is $3x + 18 = 5x + 8$ . At first glance, we can see that the equation has two variables, $x$ , and a constant term on both sides. To understand the equation better, let's rewrite it in a more simplified form.

Simplifying the Equation

To simplify the equation, we can start by subtracting $3x$ from both sides of the equation. This will help us isolate the variable $x$ on one side of the equation.

${ 3x + 18 - 3x = 5x + 8 - 3x }$

Simplifying the equation further, we get:

${ 18 = 2x + 8 }$

Now, let's subtract $8$ from both sides of the equation to get:

${ 10 = 2x }$

Understanding the Concept of One Solution

When we say that a system of equations has one solution, it means that there is only one set of values for the variables that satisfies both equations. In other words, the system of equations has a unique solution.

To understand this concept better, let's consider an analogy. Imagine that you are trying to find a specific book in a library. If the library has only one book that matches the description, then you have found the unique solution. However, if the library has multiple books that match the description, then you have found multiple solutions.

Solving the Equation

Now that we have simplified the equation, let's solve for $x$ . To do this, we can divide both sides of the equation by $2$ .

${ \frac{10}{2} = \frac{2x}{2} }$

Simplifying the equation further, we get:

${ 5 = x }$

Therefore, the solution to the equation is $x = 5$ .

Conclusion

In conclusion, we have analyzed the equation $3x + 18 = 5x + 8$ and determined that it has one solution. We have also understood the concept of one solution and how it applies to systems of equations. Finally, we have provided a step-by-step solution to the equation, which resulted in the solution $x = 5$ .

Key Takeaways

A system of equations with one solution means that there is only one set of values for the variables that satisfies both equations.
To solve a system of equations with one solution, we can use algebraic methods, such as substitution and elimination.
The solution to the equation $3x + 18 = 5x + 8$ is $x = 5$ .

Introduction

In our previous article, we explored the concept of systems of equations with one solution, specifically the equation $3x + 18 = 5x + 8$ . We analyzed the equation, determined what it means to have one solution, and provided a step-by-step solution to the equation. In this article, we will answer some frequently asked questions (FAQs) about systems of equations with one solution.

Q: What is a system of equations with one solution?

A: A system of equations with one solution means that there is only one set of values for the variables that satisfies both equations. In other words, the system of equations has a unique solution.

Q: How do I know if a system of equations has one solution?

A: To determine if a system of equations has one solution, you can use the following methods:

Check if the two equations are equivalent. If they are, then the system of equations has one solution.
Check if the two equations are parallel. If they are, then the system of equations has no solution.
Check if the two equations intersect at a single point. If they do, then the system of equations has one solution.

Q: How do I solve a system of equations with one solution?

A: To solve a system of equations with one solution, you can use algebraic methods, such as substitution and elimination. Here are the steps:

Simplify the equations by combining like terms.
Use substitution or elimination to isolate one of the variables.
Solve for the isolated variable.
Substitute the value of the isolated variable into one of the original equations to solve for the other variable.

Q: What is the difference between a system of equations with one solution and a system of equations with no solution?

A: A system of equations with one solution means that there is only one set of values for the variables that satisfies both equations. On the other hand, a system of equations with no solution means that there are no values for the variables that satisfy both equations.

Q: Can a system of equations have more than one solution?

A: No, a system of equations cannot have more than one solution. If a system of equations has more than one solution, then it is not a system of equations with one solution.

Q: How do I determine if a system of equations has infinitely many solutions?

A: To determine if a system of equations has infinitely many solutions, you can check if the two equations are equivalent. If they are, then the system of equations has infinitely many solutions.

Q: Can I use a graphing calculator to solve a system of equations with one solution?

A: Yes, you can use a graphing calculator to solve a system of equations with one solution. Simply graph the two equations on the same coordinate plane and find the point of intersection.

Conclusion

In conclusion, we have answered some frequently asked questions (FAQs) about systems of equations with one solution. We hope that this article has provided you with a better understanding of this concept and how to solve systems of equations with one solution.

Key Takeaways

A system of equations with one solution means that there is only one set of values for the variables that satisfies both equations.
To solve a system of equations with one solution, you can use algebraic methods, such as substitution and elimination.
A system of equations with one solution can be solved using a graphing calculator.