Which Graph Represents The Solution Set To This System Of Equations?${ Y = -\frac{1}{2}x + 3 } A N D And An D { Y = \frac{1}{2}x - 1 \}

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 dealing with a system of linear equations, we can use various methods to find the solution set, including graphing, substitution, and elimination. In this article, we will focus on graphing and determine which graph represents the solution set to the given system of equations.

The System of Equations

The given system of equations is:

${ y = -\frac{1}{2}x + 3 }$

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

To find the solution set, we need to find the point(s) of intersection between the two lines represented by these equations.

Graphing the Equations

Let's start by graphing the two equations on a coordinate plane.

Graph 1: y = -\frac{1}{2}x + 3

To graph this equation, we can use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is -\frac{1}{2} and the y-intercept is 3.

import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10, 400)

y1 = -0.5 * x + 3

plt.plot(x, y1, label='y = -\frac1}{2}x + 3')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph 1 y = -\frac{1{2}x + 3')
plt.legend()
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Graph 2: y = \frac{1}{2}x - 1

To graph this equation, we can use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is \frac{1}{2} and the y-intercept is -1.

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-10, 10, 400)

y2 = 0.5 * x - 1

plt.plot(x, y2, label='y = \frac1}{2}x - 1')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph 2 y = \frac{1{2}x - 1')
plt.legend()
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Finding the Solution Set

To find the solution set, we need to find the point(s) of intersection between the two lines. We can do this by setting the two equations equal to each other and solving for x.

${ -\frac{1}{2}x + 3 = \frac{1}{2}x - 1 }$

Solving for x, we get:

${ -x + 3 = x - 1 }$

${ -2x = -4 }$

${ x = 2 }$

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y.

${ y = -\frac{1}{2}x + 3 }$

${ y = -\frac{1}{2}(2) + 3 }$

${ y = -1 + 3 }$

${ y = 2 }$

Therefore, the solution set is the point (2, 2).

Which Graph Represents the Solution Set?

Based on the graphing and solving steps above, we can determine which graph represents the solution set.

The graph that represents the solution set is the one that passes through the point (2, 2).

Conclusion

In this article, we discussed how to find the solution set to a system of linear equations using graphing. We graphed the two equations, found the point(s) of intersection, and determined which graph represents the solution set. The solution set is the point (2, 2), and the graph that represents this solution set is the one that passes through this point.

Final Answer

The final answer is:

Introduction

In our previous article, we discussed how to find the solution set to a system of linear equations using graphing. We graphed the two equations, found the point(s) of intersection, and determined which graph represents the solution set. In this article, we will answer some frequently asked questions related to system of equations and graphing.

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: How do I graph a system of equations?

A: To graph a system of equations, you can use the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. You can also use the point-slope form, y - y1 = m(x - x1), where (x1, y1) is a point on the line.

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

A: To find the solution set to a system of equations, you can graph the two equations and find the point(s) of intersection. You can also use the substitution method or the elimination method to solve the system of equations.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of equations by substituting one equation into the other equation. For example, if we have the system of equations:

${ y = 2x + 3 }$

${ y = x - 2 }$

We can substitute the second equation into the first equation to get:

${ 2x + 3 = x - 2 }$

Solving for x, we get:

${ x = -5 }$

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of equations by adding or subtracting the two equations to eliminate one of the variables. For example, if we have the system of equations:

${ 2x + 3y = 7 }$

${ x - 2y = -3 }$

We can add the two equations to eliminate the variable x:

${ 3x + y = 4 }$

Solving for y, we get:

${ y = 4 - 3x }$

Now that we have the value of y, we can substitute it into one of the original equations to find the value of x.

Q: What is the point-slope form?

A: The point-slope form is a form of a linear equation that is used to graph a line. It is given by the equation:

${ y - y1 = m(x - x1) }$

where (x1, y1) is a point on the line and m is the slope.

Q: How do I determine which graph represents the solution set?

A: To determine which graph represents the solution set, you can graph the two equations and find the point(s) of intersection. The graph that passes through the point(s) of intersection represents the solution set.

Conclusion

In this article, we answered some frequently asked questions related to system of equations and graphing. We discussed the substitution method, the elimination method, and the point-slope form. We also provided examples of how to use these methods to solve a system of equations.

Final Answer

The final answer is:

• The substitution method is a method of solving a system of equations by substituting one equation into the other equation.
• The elimination method is a method of solving a system of equations by adding or subtracting the two equations to eliminate one of the variables.
• The point-slope form is a form of a linear equation that is used to graph a line.
• The graph that passes through the point(s) of intersection represents the solution set.