Solve The Following System Of Equations Graphically:${ \begin{align} y &= \frac{5}{4}x - 6 \ y &= -\frac{1}{2}x + 8 \end{align} }$Plot The Two Lines On The Provided Set Of Axes. Click A Line To Delete It If Needed.

Mar 10, 2025 by ADMIN 218 views

**Solving a System of Linear Equations Graphically**

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations graphically involves plotting the equations on a coordinate plane and finding the point of intersection, which represents the solution to the system. In this article, we will explore how to solve a system of linear equations graphically using the given equations: $y = \frac{5}{4}x - 6$ and $y = -\frac{1}{2}x + 8$ .

Understanding the Equations

Before we proceed with solving the system of equations graphically, let's understand the given equations. The first equation is $y = \frac{5}{4}x - 6$ , which represents a line with a slope of $\frac{5}{4}$ and a y-intercept of $-6$ . The second equation is $y = -\frac{1}{2}x + 8$ , which represents a line with a slope of $-\frac{1}{2}$ and a y-intercept of $8$ .

Plotting the Equations

To solve the system of equations graphically, we need to plot the two lines on a coordinate plane. We can use a graphing calculator or a computer program to plot the lines. Alternatively, we can use a piece of graph paper and a pencil to plot the lines by hand.

Plotting the First Line

To plot the first line, $y = \frac{5}{4}x - 6$ , we need to find two points on the line. We can do this by substituting different values of $x$ into the equation and solving for $y$ . Let's find two points on the line by substituting $x = 0$ and $x = 4$ into the equation.

For $x = 0$ , we have $y = \frac{5}{4}(0) - 6 = -6$ . So, the point $(0, -6)$ is on the line.
For $x = 4$ , we have $y = \frac{5}{4}(4) - 6 = 1$ . So, the point $(4, 1)$ is on the line.

Now that we have two points on the line, we can plot the line by drawing a straight line through the two points.

Plotting the Second Line

To plot the second line, $y = -\frac{1}{2}x + 8$ , we need to find two points on the line. We can do this by substituting different values of $x$ into the equation and solving for $y$ . Let's find two points on the line by substituting $x = 0$ and $x = 4$ into the equation.

For $x = 0$ , we have $y = -\frac{1}{2}(0) + 8 = 8$ . So, the point $(0, 8)$ is on the line.
For $x = 4$ , we have $y = -\frac{1}{2}(4) + 8 = 4$ . So, the point $(4, 4)$ is on the line.

Now that we have two points on the line, we can plot the line by drawing a straight line through the two points.

Finding the Point of Intersection

To find the point of intersection, we need to find the point where the two lines intersect. We can do this by finding the point where the two lines have the same $x$ -coordinate and the same $y$ -coordinate.

Let's find the point of intersection by equating the two equations:

\frac{5}{4}x - 6 = -\frac{1}{2}x + 8

Now, let's solve for $x$ :

\frac{5}{4}x + \frac{1}{2}x = 8 + 6

\frac{9}{8}x = 14

x = \frac{14 \times 8}{9}

x = \frac{112}{9}

Now that we have the value of $x$ , we can substitute it into one of the equations to find the value of $y$ . Let's substitute $x = \frac{112}{9}$ into the first equation:

y = \frac{5}{4}\left(\frac{112}{9}\right) - 6

y = \frac{140}{9} - 6

y = \frac{140 - 54}{9}

y = \frac{86}{9}

So, the point of intersection is $\left(\frac{112}{9}, \frac{86}{9}\right)$ .

Conclusion

In this article, we have explored how to solve a system of linear equations graphically using the given equations: $y = \frac{5}{4}x - 6$ and $y = -\frac{1}{2}x + 8$ . We have plotted the two lines on a coordinate plane and found the point of intersection, which represents the solution to the system. We have also discussed the importance of understanding the equations and plotting the lines accurately to find the point of intersection.

Tips and Variations

To solve a system of linear equations graphically, it is essential to plot the lines accurately and find the point of intersection.
The point of intersection represents the solution to the system of linear equations.
To find the point of intersection, we can equate the two equations and solve for $x$ and $y$ .
We can use a graphing calculator or a computer program to plot the lines and find the point of intersection.
We can also use a piece of graph paper and a pencil to plot the lines by hand.

Common Mistakes

Plotting the lines incorrectly can lead to incorrect solutions.
Failing to find the point of intersection can lead to incorrect solutions.
Equating the two equations incorrectly can lead to incorrect solutions.

Real-World Applications

Solving a system of linear equations graphically has numerous real-world applications, such as finding the intersection of two lines in a coordinate plane.
It is used in various fields, such as physics, engineering, and economics.
It is also used in computer graphics and game development.

Conclusion

Introduction

In our previous article, we explored how to solve a system of linear equations graphically using the given equations: $y = \frac{5}{4}x - 6$ and $y = -\frac{1}{2}x + 8$ . We plotted the two lines on a coordinate plane and found the point of intersection, which represents the solution to the system. In this article, we will answer some frequently asked questions about solving a system of linear equations graphically.

Q: What is the purpose of solving a system of linear equations graphically?

A: The purpose of solving a system of linear equations graphically is to find the point of intersection, which represents the solution to the system. This is useful in various fields, such as physics, engineering, and economics.

Q: How do I plot the lines on a coordinate plane?

A: To plot the lines on a coordinate plane, you need to find two points on each line. You can do this by substituting different values of $x$ into the equation and solving for $y$ . Once you have two points on each line, you can plot the lines by drawing a straight line through the two points.

Q: How do I find the point of intersection?

A: To find the point of intersection, you need to equate the two equations and solve for $x$ and $y$ . Once you have the value of $x$ , you can substitute it into one of the equations to find the value of $y$ .

Q: What are some common mistakes to avoid when solving a system of linear equations graphically?

A: Some common mistakes to avoid when solving a system of linear equations graphically include:

Plotting the lines incorrectly
Failing to find the point of intersection
Equating the two equations incorrectly

Q: How do I use a graphing calculator or computer program to solve a system of linear equations graphically?

A: To use a graphing calculator or computer program to solve a system of linear equations graphically, you need to enter the equations into the calculator or program and plot the lines. The calculator or program will then find the point of intersection, which represents the solution to the system.

Q: Can I use a piece of graph paper and a pencil to solve a system of linear equations graphically?

A: Yes, you can use a piece of graph paper and a pencil to solve a system of linear equations graphically. This is a good way to practice plotting lines and finding the point of intersection.

Q: How do I know if the point of intersection is the correct solution to the system?

A: To know if the point of intersection is the correct solution to the system, you need to check that the point of intersection satisfies both equations. If it does, then it is the correct solution to the system.

Q: Can I use solving a system of linear equations graphically to solve a system of more than two equations?

A: Yes, you can use solving a system of linear equations graphically to solve a system of more than two equations. However, this can be more complicated and may require the use of a graphing calculator or computer program.

Q: What are some real-world applications of solving a system of linear equations graphically?

A: Some real-world applications of solving a system of linear equations graphically include:

Finding the intersection of two lines in a coordinate plane
Solving systems of linear equations in physics, engineering, and economics
Using computer graphics and game development

Conclusion

In conclusion, solving a system of linear equations graphically is a powerful tool that can be used to find the intersection of two lines in a coordinate plane. It has numerous real-world applications and is used in various fields. By understanding the equations and plotting the lines accurately, we can find the point of intersection and solve the system of linear equations.