Solve The Following System Of Equations By Graphing. Then Determine Whether The System Is Consistent Or Inconsistent, And Whether The Equations Are Dependent Or Independent. If The System Is Consistent, Give The Solution.$[ \begin{cases} 4x + 2y

Introduction

In this article, we will explore the process of solving a system of linear equations using the graphing method. We will also determine whether the system is consistent or inconsistent, and whether the equations are dependent or independent. If the system is consistent, we will provide the solution.

What is a System of Equations?

A system of equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. In this case, we have two linear equations:

${ 4x + 2y = 12 }$ ${ 2x + y = 6 }$

Graphing the Equations

To solve the system of equations by graphing, we need to graph each equation on the same coordinate plane. 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.

Graphing the First Equation

The first equation is:

${ 4x + 2y = 12 }$

We can rewrite this equation in slope-intercept form as:

${ y = -2x + 6 }$

This equation has a slope of -2 and a y-intercept of 6. We can graph this equation by plotting two points on the coordinate plane and drawing a line through them.

Graphing the Second Equation

The second equation is:

${ 2x + y = 6 }$

We can rewrite this equation in slope-intercept form as:

${ y = -2x + 6 }$

This equation has the same slope as the first equation, but a different y-intercept. We can graph this equation by plotting two points on the coordinate plane and drawing a line through them.

Graphing the System of Equations

Now that we have graphed both equations, we can graph the system of equations by drawing a line through the points of intersection of the two graphs.

Determining Consistency and Independence

To determine whether the system is consistent or inconsistent, we need to check if the two graphs intersect at any point. If they intersect, the system is consistent. If they do not intersect, the system is inconsistent.

To determine whether the equations are dependent or independent, we need to check if the two graphs are the same or if they are parallel. If the graphs are the same, the equations are dependent. If the graphs are parallel, the equations are independent.

Solving the System of Equations

If the system is consistent, we can solve for the values of x and y by finding the point of intersection of the two graphs.

Solution

To find the point of intersection, we can set the two equations equal to each other and solve for x and y.

${ 4x + 2y = 12 }$ ${ 2x + y = 6 }$

We can multiply the second equation by 2 to get:

${ 4x + 2y = 12 }$ ${ 4x + 2y = 12 }$

Now we can subtract the second equation from the first equation to get:

${ 0 = 0 }$

This means that the two equations are the same, and the system is consistent. We can solve for x and y by substituting the value of x into one of the equations.

Conclusion

In this article, we have solved a system of linear equations using the graphing method. We have determined whether the system is consistent or inconsistent, and whether the equations are dependent or independent. If the system is consistent, we have provided the solution.

Key Takeaways

• A system of equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.
• To solve a system of equations by graphing, we need to graph each equation on the same coordinate plane.
• To determine whether the system is consistent or inconsistent, we need to check if the two graphs intersect at any point.
• To determine whether the equations are dependent or independent, we need to check if the two graphs are the same or if they are parallel.
• If the system is consistent, we can solve for the values of x and y by finding the point of intersection of the two graphs.

Final Answer

The final answer is:

${ x = 3 }$ ${ y = 3 }$

Introduction

In our previous article, we explored the process of solving a system of linear equations using the graphing method. We determined whether the system is consistent or inconsistent, and whether the equations are dependent or independent. If the system is consistent, we provided the solution.

In this article, we will answer some frequently asked questions about solving a system of equations by graphing.

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

A: The first step in solving a system of equations by graphing is to graph each equation on the same coordinate plane. 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.

Q: How do I determine whether the system is consistent or inconsistent?

A: To determine whether the system is consistent or inconsistent, we need to check if the two graphs intersect at any point. If they intersect, the system is consistent. If they do not intersect, the system is inconsistent.

Q: How do I determine whether the equations are dependent or independent?

A: To determine whether the equations are dependent or independent, we need to check if the two graphs are the same or if they are parallel. If the graphs are the same, the equations are dependent. If the graphs are parallel, the equations are independent.

Q: What if the system is inconsistent? What does this mean?

A: If the system is inconsistent, it means that the two equations are contradictory, and there is no solution. This can happen when the two equations represent parallel lines that never intersect.

Q: What if the system is consistent, but the equations are dependent? What does this mean?

A: If the system is consistent, but the equations are dependent, it means that the two equations represent the same line. In this case, there are infinitely many solutions, and we can write the solution as an equation.

Q: How do I find the solution to a system of equations if it is consistent?

A: If the system is consistent, we can find the solution by finding the point of intersection of the two graphs. We can do this by setting the two equations equal to each other and solving for x and y.

Q: What if I am having trouble graphing the equations? What can I do?

A: If you are having trouble graphing the equations, try the following:

• Use a graphing calculator to graph the equations.
• Use a graphing software to graph the equations.
• Plot two points on the coordinate plane and draw a line through them.
• Use the slope-intercept form of a linear equation to graph the equations.

Q: Can I use other methods to solve a system of equations besides graphing?

A: Yes, there are other methods to solve a system of equations besides graphing. Some of these methods include:

• Substitution method
• Elimination method
• Matrix method
• Cramer's rule

Conclusion

In this article, we have answered some frequently asked questions about solving a system of equations by graphing. We have discussed the first step in solving a system of equations by graphing, how to determine whether the system is consistent or inconsistent, and how to determine whether the equations are dependent or independent. We have also discussed what to do if the system is inconsistent or if the equations are dependent.

Key Takeaways

• The first step in solving a system of equations by graphing is to graph each equation on the same coordinate plane.
• To determine whether the system is consistent or inconsistent, we need to check if the two graphs intersect at any point.
• To determine whether the equations are dependent or independent, we need to check if the two graphs are the same or if they are parallel.
• If the system is consistent, we can find the solution by finding the point of intersection of the two graphs.
• There are other methods to solve a system of equations besides graphing.

Final Answer

The final answer is:

• The first step in solving a system of equations by graphing is to graph each equation on the same coordinate plane.
• To determine whether the system is consistent or inconsistent, we need to check if the two graphs intersect at any point.
• To determine whether the equations are dependent or independent, we need to check if the two graphs are the same or if they are parallel.
• If the system is consistent, we can find the solution by finding the point of intersection of the two graphs.
• There are other methods to solve a system of equations besides graphing.