Solve The System Of Equations By Graphing. Check Your Solution.$\[ \begin{array}{l} 4x - 5y = 20 \\ 6x + Y = -4 \end{array} \\]Use The Graphing Tool To Graph The System.

Feb 28, 2025 by ADMIN 170 views

**Solving Systems of Equations by Graphing: A Step-by-Step Guide**

Introduction

Solving systems of equations is a fundamental concept in mathematics, and there are several methods to solve them. One of the most effective methods is graphing, which involves plotting the equations on a coordinate plane and finding the point of intersection. In this article, we will explore how to solve a system of equations by graphing, using the given system as an example.

Understanding the System of Equations

The given system of equations is:

{ \begin{array}{l} 4x - 5y = 20 \\ 6x + y = -4 \end{array} \}

To solve this system, we need to find the values of x and y that satisfy both equations simultaneously.

Graphing the First Equation

The first equation is $4x - 5y = 20$ . To graph this equation, we can use the slope-intercept form, which is $y = mx + b$ , where m is the slope and b is the y-intercept.

To convert the equation to slope-intercept form, we can solve for y:

$4x - 5y = 20$

$-5y = -4x + 20$

$y = \frac{4}{5}x - 4$

Now that we have the equation in slope-intercept form, we can graph it on a coordinate plane. The graph of this equation is a line with a slope of $\frac{4}{5}$ and a y-intercept of -4.

Graphing the Second Equation

The second equation is $6x + y = -4$ . To graph this equation, we can also use the slope-intercept form.

To convert the equation to slope-intercept form, we can solve for y:

$6x + y = -4$

$y = -6x - 4$

Now that we have the equation in slope-intercept form, we can graph it on a coordinate plane. The graph of this equation is a line with a slope of -6 and a y-intercept of -4.

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 x-coordinate of the point of intersection and then substituting it into one of the equations to find the y-coordinate.

Let's find the x-coordinate of the point of intersection. We can do this by setting the two equations equal to each other:

$4x - 5y = 20$

$6x + y = -4$

We can solve for x by multiplying the first equation by 6 and the second equation by 4:

$24x - 30y = 120$

$24x + 4y = -16$

Now we can add the two equations together to eliminate the y-variable:

$48x - 26y = 104$

Now we can solve for x:

$48x = 104 + 26y$

$x = \frac{104 + 26y}{48}$

Now that we have the x-coordinate of the point of intersection, we can substitute it into one of the equations to find the y-coordinate. Let's substitute it into the first equation:

$4x - 5y = 20$

$4(\frac{104 + 26y}{48}) - 5y = 20$

Now we can simplify the equation:

$\frac{104 + 26y}{12} - 5y = 20$

Now we can multiply both sides of the equation by 12 to eliminate the fraction:

$104 + 26y - 60y = 240$

Now we can combine like terms:

$-34y = 136$

Now we can solve for y:

$y = -4$

Now that we have the y-coordinate of the point of intersection, we can substitute it into one of the equations to find the x-coordinate. Let's substitute it into the first equation:

$4x - 5y = 20$

$4x - 5(-4) = 20$

Now we can simplify the equation:

$4x + 20 = 20$

Now we can solve for x:

$4x = 0$

$x = 0$

Conclusion

In this article, we have learned how to solve a system of equations by graphing. We have used the given system as an example and have found the point of intersection by graphing the two equations on a coordinate plane. We have also used the slope-intercept form to graph the equations and have found the x and y-coordinates of the point of intersection.

Discussion

What are some other methods for solving systems of equations?
How can you use graphing to solve systems of equations with more than two variables?
What are some real-world applications of solving systems of equations?

Additional Resources

Khan Academy: Solving Systems of Equations
Mathway: Solving Systems of Equations
Wolfram Alpha: Solving Systems of Equations

Final Thoughts

Introduction

In our previous article, we explored how to solve systems of equations by graphing. We used the given system as an example and found the point of intersection by graphing the two equations on a coordinate plane. In this article, we will answer some frequently asked questions about solving systems of equations by graphing.

Q: What are some common mistakes to avoid when graphing systems of equations?

A: When graphing systems of equations, some common mistakes to avoid include:

Not using a ruler or straightedge to draw the lines
Not labeling the axes and the point of intersection
Not using a consistent scale for the graph
Not checking for extraneous solutions

Q: How can I determine if a system of equations has a unique solution, no solution, or infinitely many solutions?

A: To determine if a system of equations has a unique solution, no solution, or infinitely many solutions, you can use the following methods:

Graph the two equations on a coordinate plane and count the number of points of intersection. If there is one point of intersection, the system has a unique solution. If there are no points of intersection, the system has no solution. If there are infinitely many points of intersection, the system has infinitely many solutions.
Use the slope-intercept form to graph the equations. If the lines are parallel, the system has no solution. If the lines intersect at a single point, the system has a unique solution. If the lines are coincident, the system has infinitely many solutions.

Q: How can I use graphing to solve systems of equations with more than two variables?

A: To use graphing to solve systems of equations with more than two variables, you can use the following methods:

Use a three-dimensional coordinate system to graph the equations.
Use a parametric equation to graph the equations.
Use a computer algebra system to graph the equations.

Q: What are some real-world applications of solving systems of equations?

A: Some real-world applications of solving systems of equations include:

Physics: Solving systems of equations is used to model the motion of objects in physics.
Engineering: Solving systems of equations is used to design and optimize systems in engineering.
Economics: Solving systems of equations is used to model economic systems and make predictions about the economy.
Computer Science: Solving systems of equations is used in computer science to solve problems in computer graphics, game development, and artificial intelligence.

Q: How can I use graphing to solve systems of equations with rational coefficients?

A: To use graphing to solve systems of equations with rational coefficients, you can use the following methods:

Use the rational root theorem to find the rational roots of the equations.
Use synthetic division to divide the equations by the rational roots.
Use the quadratic formula to solve the resulting quadratic equations.

Q: What are some tips for graphing systems of equations?

A: Some tips for graphing systems of equations include:

Use a ruler or straightedge to draw the lines.
Label the axes and the point of intersection.
Use a consistent scale for the graph.
Check for extraneous solutions.
Use a computer algebra system to graph the equations.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of equations by graphing. We have discussed common mistakes to avoid, how to determine if a system of equations has a unique solution, no solution, or infinitely many solutions, and how to use graphing to solve systems of equations with more than two variables. We have also discussed real-world applications of solving systems of equations and provided some tips for graphing systems of equations.

Discussion

What are some other methods for solving systems of equations?
How can you use graphing to solve systems of equations with more than two variables?
What are some real-world applications of solving systems of equations?

Additional Resources

Khan Academy: Solving Systems of Equations
Mathway: Solving Systems of Equations
Wolfram Alpha: Solving Systems of Equations

Final Thoughts

Solving systems of equations is a fundamental concept in mathematics, and graphing is one of the most effective methods for solving them. By graphing the equations on a coordinate plane and finding the point of intersection, we can find the values of x and y that satisfy both equations simultaneously. We hope that this article has provided you with a better understanding of how to solve systems of equations by graphing.