The Solution To The System Below Is $(1, -3)$.1. $2y + X = -5$2. Y + 3 X = 0 Y + 3x = 0 Y + 3 X = 0 Which Graph Correctly Represents This System?

Introduction

Solving a system of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. A system of linear equations consists of two or more linear equations that involve the same set of variables. In this article, we will explore how to represent a system of linear equations graphically and determine which graph correctly represents the given system.

Understanding the System of Linear Equations

The given system of linear equations is:

1. $2y + x = -5$
2. $y + 3x = 0$

To solve this system, we can use the method of substitution or elimination. However, in this article, we will focus on representing the system graphically.

Graphing the First Equation

The first equation is $2y + x = -5$. To graph this equation, we can rewrite it in the slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Rearranging the equation, we get:

$2y = -x - 5$

Dividing both sides by 2, we get:

$y = -\frac{1}{2}x - \frac{5}{2}$

This is the slope-intercept form of the first equation. The slope is $-\frac{1}{2}$, and the y-intercept is $-\frac{5}{2}$.

Graphing the Second Equation

The second equation is $y + 3x = 0$. To graph this equation, we can rewrite it in the slope-intercept form.

Rearranging the equation, we get:

$y = -3x$

This is the slope-intercept form of the second equation. The slope is $-3$, and the y-intercept is $0$.

Graphing the System

To graph the system, we need to graph both equations on the same coordinate plane. The graph of the first equation is a line with a slope of $-\frac{1}{2}$ and a y-intercept of $-\frac{5}{2}$. The graph of the second equation is a line with a slope of $-3$ and a y-intercept of $0$.

Determining the Correct Graph

To determine which graph correctly represents the system, we need to find the point of intersection between the two lines. The point of intersection is the solution to the system.

Using the method of substitution, we can substitute the expression for $y$ from the second equation into the first equation.

$2y + x = -5$

Substituting $y = -3x$, we get:

$2(-3x) + x = -5$

Simplifying the equation, we get:

$-6x + x = -5$

Combine like terms:

$-5x = -5$

Dividing both sides by $-5$, we get:

$x = 1$

Substituting $x = 1$ into the second equation, we get:

$y + 3(1) = 0$

Simplifying the equation, we get:

$y + 3 = 0$

Subtracting 3 from both sides, we get:

$y = -3$

Therefore, the solution to the system is $(1, -3)$.

Conclusion

In this article, we have explored how to represent a system of linear equations graphically and determine which graph correctly represents the given system. We have graphed the first equation, graphed the second equation, and determined the point of intersection between the two lines. The point of intersection is the solution to the system, which is $(1, -3)$. This solution satisfies both equations in the system.

Graphs of the System

Here are the graphs of the system:

Graph 1

This graph represents the first equation, $2y + x = -5$. The graph is a line with a slope of $-\frac{1}{2}$ and a y-intercept of $-\frac{5}{2}$.

Graph 2

This graph represents the second equation, $y + 3x = 0$. The graph is a line with a slope of $-3$ and a y-intercept of $0$.

Graph 3

This graph represents the system of linear equations. The graph is a line with a slope of $-\frac{1}{2}$ and a y-intercept of $-\frac{5}{2}$, and it intersects with the line representing the second equation at the point $(1, -3)$.

Which Graph Correctly Represents the System?

Based on the solution to the system, which is $(1, -3)$, we can determine that the correct graph is Graph 3. This graph represents the system of linear equations and intersects with the line representing the second equation at the point $(1, -3)$.

Final Answer

The final answer is Graph 3.

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, economics, and computer science. In this article, we will answer some frequently asked questions (FAQs) about solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. For example:

1. $2y + x = -5$
2. $y + 3x = 0$

Q: How do I solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including:

1. Substitution Method: Substitute the expression for one variable from one equation into the other equation.
2. Elimination Method: Add or subtract the equations to eliminate one variable.
3. Graphical Method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the point of intersection?

A: The point of intersection is the solution to the system of linear equations. It is the point where the two lines intersect on the coordinate plane.

Q: How do I find the point of intersection?

A: To find the point of intersection, you can use the substitution method or the elimination method. Alternatively, you can graph the equations on a coordinate plane and find the point where the two lines intersect.

Q: What is the solution to the system of linear equations?

A: The solution to the system of linear equations is the point of intersection, which is the point where the two lines intersect on the coordinate plane.

Q: How do I determine which graph correctly represents the system of linear equations?

A: To determine which graph correctly represents the system of linear equations, you need to find the point of intersection between the two lines. The graph that intersects at the point of intersection is the correct graph.

Q: What is the significance of the solution to the system of linear equations?

A: The solution to the system of linear equations represents the values of the variables that satisfy both equations in the system. It is a critical concept in mathematics and has numerous applications in various fields.

Q: Can I use technology to solve systems of linear equations?

A: Yes, you can use technology such as graphing calculators or computer software to solve systems of linear equations. These tools can help you graph the equations and find the point of intersection.

Q: Are there any limitations to solving systems of linear equations?

A: Yes, there are limitations to solving systems of linear equations. For example, if the system has no solution or an infinite number of solutions, it may not be possible to find a unique solution.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about solving systems of linear equations. We have discussed the definition of a system of linear equations, the methods for solving it, and the significance of the solution. We have also addressed some common limitations and applications of solving systems of linear equations.

Additional Resources

For more information on solving systems of linear equations, you can consult the following resources:

• Textbooks: "Linear Algebra and Its Applications" by Gilbert Strang, "Introduction to Linear Algebra" by Jim Hefferon
• Online Resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Software: Graphing calculators, computer software such as Mathematica or MATLAB

Final Answer

The final answer is that solving systems of linear equations is a fundamental concept in mathematics with numerous applications in various fields. It requires a deep understanding of linear algebra and the ability to use various methods to solve the system.