Examine The System Of Equations.$\[ \begin{array}{l} y = 2x - 3 \\ y = -3 \end{array} \\]Which Statement About The System Of Linear Equations Is True?A. The Lines Have Different Slopes.B. There Is No Solution To The System.C. The Lines Have

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. In this article, we will examine a system of linear equations and determine which statement about the system is true.

The System of Equations

The system of equations we will be examining is given by:

$$\begin{array}{l} y = 2x - 3 \\ y = -3 \end{array}$$

Understanding the Equations

The first equation, $y = 2x - 3$, is a linear equation in two variables, $x$ and $y$. It represents a line in the $xy$-plane. The second equation, $y = -3$, is also a linear equation in two variables, $x$ and $y$. However, it represents a horizontal line in the $xy$-plane.

Analyzing the System

To analyze the system, we need to determine the relationship between the two equations. We can do this by setting the two equations equal to each other and solving for $x$.

$$2x - 3 = -3$$

Solving for $x$, we get:

$$2x = 0$$

$$x = 0$$

Now that we have found the value of $x$, we can substitute it into either of the original equations to find the value of $y$. Let's substitute $x = 0$ into the first equation:

$$y = 2(0) - 3$$

$$y = -3$$

Conclusion

We have found that the system of equations has a solution, $x = 0$ and $y = -3$. However, this solution is not unique, as there are infinitely many solutions to the system. This is because the second equation, $y = -3$, is a horizontal line that intersects the first equation at every point where $y = -3$.

Which Statement is True?

Now that we have analyzed the system, we can determine which statement about the system is true.

A. The lines have different slopes.

• The first equation, $y = 2x - 3$, has a slope of 2.
• The second equation, $y = -3$, has a slope of 0 (since it is a horizontal line).
• Therefore, the lines have different slopes.

B. There is no solution to the system.

• We have found that the system has a solution, $x = 0$ and $y = -3$.
• Therefore, this statement is false.

C. The lines have the same slope.

• We have found that the first equation has a slope of 2, while the second equation has a slope of 0.
• Therefore, the lines do not have the same slope.

Conclusion

In conclusion, the statement that is true about the system of equations is:

• A. The lines have different slopes.

This is because the first equation has a slope of 2, while the second equation has a slope of 0. Therefore, the lines have different slopes.

Final Answer

The final answer is A. The lines have different slopes.

Additional Information

• A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
• A linear equation in two variables is an equation that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• A horizontal line is a line that has a slope of 0.
• A system of equations can have a unique solution, infinitely many solutions, or no solution.

References

• [1] "Systems of Equations" by Math Open Reference
• [2] "Linear Equations" by Math Is Fun
• [3] "Horizontal Lines" by Purplemath
  Examine the System of Equations: Q&A =====================================

Introduction

In our previous article, we examined a system of linear equations and determined that the statement "The lines have different slopes" is true. In this article, we will answer some frequently asked questions about the system of equations.

Q&A

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: What is a linear equation in two variables?

A: A linear equation in two variables is an equation that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is a horizontal line?

A: A horizontal line is a line that has a slope of 0.

Q: How do you analyze a system of equations?

A: To analyze a system of equations, you need to determine the relationship between the two equations. You can do this by setting the two equations equal to each other and solving for $x$.

Q: What is the difference between a system of equations with a unique solution, infinitely many solutions, and no solution?

A: A system of equations with a unique solution has only one solution. A system of equations with infinitely many solutions has an infinite number of solutions. A system of equations with no solution has no solution.

Q: How do you determine the number of solutions to a system of equations?

A: To determine the number of solutions to a system of equations, you need to analyze the system and determine the relationship between the two equations.

Q: What is the significance of the slope of a line in a system of equations?

A: The slope of a line in a system of equations determines the relationship between the two equations. If the slopes are different, the lines intersect at a single point. If the slopes are the same, the lines are parallel and do not intersect.

Q: Can a system of equations have a solution if the lines are parallel?

A: No, a system of equations cannot have a solution if the lines are parallel.

Q: Can a system of equations have a solution if the lines intersect at a single point?

A: Yes, a system of equations can have a solution if the lines intersect at a single point.

Q: How do you find the solution to a system of equations?

A: To find the solution to a system of equations, you need to analyze the system and determine the relationship between the two equations. You can then use algebraic methods to solve for the variables.

Q: What is the importance of solving systems of equations?

A: Solving systems of equations is important in many fields, including mathematics, science, and engineering. It is used to model real-world problems and make predictions.

Conclusion

In conclusion, solving systems of equations is an important skill that is used in many fields. By understanding the concepts of linear equations, horizontal lines, and the relationship between the two equations, you can analyze a system of equations and determine the number of solutions.

Final Answer

The final answer is that solving systems of equations is an important skill that is used in many fields.

Additional Information

• A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
• A linear equation in two variables is an equation that can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• A horizontal line is a line that has a slope of 0.
• A system of equations can have a unique solution, infinitely many solutions, or no solution.
• The slope of a line in a system of equations determines the relationship between the two equations.
• A system of equations cannot have a solution if the lines are parallel.
• A system of equations can have a solution if the lines intersect at a single point.

References

• [1] "Systems of Equations" by Math Open Reference
• [2] "Linear Equations" by Math Is Fun
• [3] "Horizontal Lines" by Purplemath