Solve This System Of Equations By Graphing. First, Graph The Equations And Find The Solution.${ \begin{align} 2x - Y &= 4 \ y &= -6x + 4 \end{align} }$Click To Select Points On The Graph.

Mar 13, 2025 by ADMIN 191 views

Solve this System of Equations by Graphing

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Introduction

Understanding the System of Equations

The given system of equations consists of two linear equations:

$2x - y = 4$
$y = -6x + 4$

We need to graph these equations on a coordinate plane and find the solution to the system.

Graphing the First Equation

To graph the first equation, $2x - y = 4$ , we can use the slope-intercept form of a linear equation, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

First, we need to isolate $y$ in the equation. We can do this by subtracting $2x$ from both sides of the equation:

-y = -2x + 4

Next, we can multiply both sides of the equation by $-1$ to get:

y = 2x - 4

Now, we can see that the slope of the line is $2$ and the y-intercept is $-4$ . We can use this information to graph the line.

Graphing the Second Equation

The second equation, $y = -6x + 4$ , is already in slope-intercept form. We can see that the slope of the line is $-6$ and the y-intercept is $4$ . We can use this information to graph the line.

Graphing the Equations

To graph the equations, we can use a coordinate plane. We can plot the points on the plane and draw the lines.

For the first equation, $y = 2x - 4$ , we can plot the points $(0, -4)$ and $(1, -2)$ . We can then draw a line through these points.

For the second equation, $y = -6x + 4$ , we can plot the points $(0, 4)$ and $(1, -2)$ . We can then draw a line through these points.

Finding the Solution

To find the solution to the system, we need to find the point of intersection between the two lines. We can do this by looking for the point where the two lines cross.

From the graph, we can see that the two lines intersect at the point $(1, -2)$ . This means that the solution to the system is $x = 1$ and $y = -2$ .

Conclusion

In this article, we learned how to graph the given system of equations and find the solution. We used the slope-intercept form of a linear equation to graph the lines and found the point of intersection between the two lines. The solution to the system is $x = 1$ and $y = -2$ .

Example Problems

Problem 1

Graph the system of equations:

$x + y = 2$
$y = -x + 2$

Find the solution to the system.

Solution

To graph the first equation, $x + y = 2$ , we can use the slope-intercept form of a linear equation, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

First, we need to isolate $y$ in the equation. We can do this by subtracting $x$ from both sides of the equation:

y = -x + 2

Now, we can see that the slope of the line is $-1$ and the y-intercept is $2$ . We can use this information to graph the line.

For the second equation, $y = -x + 2$ , we can plot the points $(0, 2)$ and $(1, 1)$ . We can then draw a line through these points.

To find the solution to the system, we need to find the point of intersection between the two lines. We can do this by looking for the point where the two lines cross.

From the graph, we can see that the two lines intersect at the point $(1, 1)$ . This means that the solution to the system is $x = 1$ and $y = 1$ .

Problem 2

Graph the system of equations:

$2x + y = 3$
$y = -2x + 3$

Find the solution to the system.

Solution

To graph the first equation, $2x + y = 3$ , we can use the slope-intercept form of a linear equation, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

First, we need to isolate $y$ in the equation. We can do this by subtracting $2x$ from both sides of the equation:

y = -2x + 3

Now, we can see that the slope of the line is $-2$ and the y-intercept is $3$ . We can use this information to graph the line.

For the second equation, $y = -2x + 3$ , we can plot the points $(0, 3)$ and $(1, 1)$ . We can then draw a line through these points.

To find the solution to the system, we need to find the point of intersection between the two lines. We can do this by looking for the point where the two lines cross.

From the graph, we can see that the two lines intersect at the point $(1, 1)$ . This means that the solution to the system is $x = 1$ and $y = 1$ .

Final Thoughts

Graphing is a powerful tool for solving systems of linear equations. By graphing the equations on a coordinate plane, we can visually identify the solution to the system. In this article, we learned how to graph the given system of equations and find the solution. We used the slope-intercept form of a linear equation to graph the lines and found the point of intersection between the two lines. The solution to the system is $x = 1$ and $y = -2$ .

References

Keywords

Graphing systems of equations
Solving systems of equations by graphing
Linear equations
Coordinate plane
Slope-intercept form
Point of intersection

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Introduction

In our previous article, we learned how to graph the given system of equations and find the solution. However, we know that there are many more questions and doubts that readers may have. In this article, we will address some of the most frequently asked questions about solving systems of equations by graphing.

Q&A

Q1: What is the first step in solving a system of equations by graphing?

A1: The first step in solving a system of equations by graphing is to graph the two equations on a coordinate plane.

Q2: How do I graph a linear equation in slope-intercept form?

A2: To graph a linear equation in slope-intercept form, you need to identify the slope (m) and the y-intercept (b). You can then plot the points on the coordinate plane and draw a line through them.

Q3: What is the point of intersection in a system of equations?

A3: The point of intersection in a system of equations is the point where the two lines cross. This point represents the solution to the system.

Q4: How do I find the point of intersection between two lines?

A4: To find the point of intersection between two lines, you need to look for the point where the two lines cross. You can do this by plotting the points on the coordinate plane and drawing a line through them.

Q5: What if the two lines do not intersect?

A5: If the two lines do not intersect, then the system of equations has no solution.

Q6: Can I use graphing to solve systems of equations with more than two variables?

A6: No, graphing is only used to solve systems of linear equations with two variables.

Q7: How do I determine if a system of equations has a unique solution, no solution, or infinitely many solutions?

A7: You can determine the type of solution by looking at the graph of the system. If the two lines intersect at a single point, then the system has a unique solution. If the two lines do not intersect, then the system has no solution. If the two lines are the same, then the system has infinitely many solutions.

Q8: Can I use graphing to solve systems of equations with non-linear equations?

A8: No, graphing is only used to solve systems of linear equations.

Q9: How do I graph a system of equations with fractions?

A9: To graph a system of equations with fractions, you need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q10: Can I use graphing to solve systems of equations with absolute value equations?

A10: No, graphing is not the best method for solving systems of equations with absolute value equations. You can use other methods such as substitution or elimination.

Conclusion

In this article, we addressed some of the most frequently asked questions about solving systems of equations by graphing. We hope that this article has provided you with a better understanding of how to graph systems of equations and find the solution.

Example Problems

Problem 1

Graph the system of equations:

$x + y = 2$
$y = -x + 2$

Find the solution to the system.

Solution

To graph the first equation, $x + y = 2$ , we can use the slope-intercept form of a linear equation, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

First, we need to isolate $y$ in the equation. We can do this by subtracting $x$ from both sides of the equation:

y = -x + 2

Now, we can see that the slope of the line is $-1$ and the y-intercept is $2$ . We can use this information to graph the line.

For the second equation, $y = -x + 2$ , we can plot the points $(0, 2)$ and $(1, 1)$ . We can then draw a line through these points.

To find the solution to the system, we need to find the point of intersection between the two lines. We can do this by looking for the point where the two lines cross.

From the graph, we can see that the two lines intersect at the point $(1, 1)$ . This means that the solution to the system is $x = 1$ and $y = 1$ .

Problem 2

Graph the system of equations:

$2x + y = 3$
$y = -2x + 3$

Find the solution to the system.

Solution

To graph the first equation, $2x + y = 3$ , we can use the slope-intercept form of a linear equation, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

First, we need to isolate $y$ in the equation. We can do this by subtracting $2x$ from both sides of the equation:

y = -2x + 3

Now, we can see that the slope of the line is $-2$ and the y-intercept is $3$ . We can use this information to graph the line.

For the second equation, $y = -2x + 3$ , we can plot the points $(0, 3)$ and $(1, 1)$ . We can then draw a line through these points.

To find the solution to the system, we need to find the point of intersection between the two lines. We can do this by looking for the point where the two lines cross.

From the graph, we can see that the two lines intersect at the point $(1, 1)$ . This means that the solution to the system is $x = 1$ and $y = 1$ .

Final Thoughts

References

Keywords

Graphing systems of equations
Solving systems of equations by graphing
Linear equations
Coordinate plane
Slope-intercept form
Point of intersection