Graph The Following System Of Equations:${ \begin{array}{l} 3x + 6y = 6 \ 2x + 2y = 6 \end{array} }$What Is The Solution To The System?A. There Are Infinitely Many Solutions.B. There Is One Unique Solution, { (4, -1)$}$.C. There

Introduction

Graphing and solving systems of linear equations is a fundamental concept in mathematics, particularly in algebra and geometry. It involves finding the intersection points of two or more lines on a coordinate plane. In this article, we will explore how to graph a system of linear equations and find the solution to the system.

Graphing a System of Linear Equations

To graph a system of linear equations, we need to first understand the concept of linear equations and their graphs. A linear equation is an equation in which the highest power of the variable(s) is 1. The graph of a linear equation is a straight line on a coordinate plane.

The given system of linear equations is:

{ \begin{array}{l} 3x + 6y = 6 \\ 2x + 2y = 6 \end{array} \}

To graph this system, we need to first graph each equation separately. We can do this by finding the x-intercept and y-intercept of each equation.

Graphing the First Equation

The first equation is $3x + 6y = 6$. To find the x-intercept, we set $y = 0$ and solve for $x$.

$3x + 6(0) = 6$

$3x = 6$

$x = 2$

So, the x-intercept of the first equation is $(2, 0)$.

To find the y-intercept, we set $x = 0$ and solve for $y$.

$3(0) + 6y = 6$

$6y = 6$

$y = 1$

So, the y-intercept of the first equation is $(0, 1)$.

Graphing the Second Equation

The second equation is $2x + 2y = 6$. To find the x-intercept, we set $y = 0$ and solve for $x$.

$2x + 2(0) = 6$

$2x = 6$

$x = 3$

So, the x-intercept of the second equation is $(3, 0)$.

To find the y-intercept, we set $x = 0$ and solve for $y$.

$2(0) + 2y = 6$

$2y = 6$

$y = 3$

So, the y-intercept of the second equation is $(0, 3)$.

Graphing the System

Now that we have found the x-intercept and y-intercept of each equation, we can graph the system of linear equations.

The graph of the first equation is a line that passes through the points $(2, 0)$ and $(0, 1)$.

The graph of the second equation is a line that passes through the points $(3, 0)$ and $(0, 3)$.

To graph the system, we need to find the intersection point of the two lines.

Finding the Intersection Point

To find the intersection point, we need to solve the system of linear equations.

We can use the method of substitution or elimination to solve the system.

Let's use the method of elimination.

We can multiply the first equation by 2 and the second equation by 3 to make the coefficients of $x$ equal.

$6x + 12y = 12$

$6x + 6y = 18$

Now, we can subtract the second equation from the first equation to eliminate the variable $x$.

$(6x + 12y) - (6x + 6y) = 12 - 18$

$6y = -6$

$y = -1$

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

Let's substitute $y = -1$ into the first equation.

$3x + 6(-1) = 6$

$3x - 6 = 6$

$3x = 12$

$x = 4$

So, the intersection point of the two lines is $(4, -1)$.

Conclusion

In this article, we have graphed a system of linear equations and found the solution to the system. We have used the method of elimination to solve the system and found the intersection point of the two lines.

The solution to the system is $(4, -1)$.

This means that the two lines intersect at the point $(4, -1)$.

Therefore, the correct answer is:

B. There is one unique solution, $(4, -1)$.

Discussion

Graphing and solving systems of linear equations is an important concept in mathematics. It involves finding the intersection points of two or more lines on a coordinate plane.

In this article, we have graphed a system of linear equations and found the solution to the system. We have used the method of elimination to solve the system and found the intersection point of the two lines.

The solution to the system is $(4, -1)$.

This means that the two lines intersect at the point $(4, -1)$.

Therefore, the correct answer is:

B. There is one unique solution, $(4, -1)$.

Final Answer

The final answer is:

B. There is one unique solution, $(4, -1)$.

References

• [1] "Graphing and Solving Systems of Linear Equations" by Math Open Reference
• [2] "Systems of Linear Equations" by Khan Academy
• [3] "Graphing Systems of Linear Equations" by Purplemath
  

Introduction

In our previous article, we explored how to graph a system of linear equations and find the solution to the system. We used the method of elimination to solve the system and found the intersection point of the two lines.

In this article, we will answer some frequently asked questions about graphing and solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. Each equation in the system is a linear equation, which means that the highest power of the variable(s) is 1.

Q: How do I graph a system of linear equations?

A: To graph a system of linear equations, you need to first graph each equation separately. You can do this by finding the x-intercept and y-intercept of each equation. Then, you can find the intersection point of the two lines by solving the system of linear equations.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations is a set of linear equations that are solved simultaneously. A system of nonlinear equations is a set of nonlinear equations that are solved simultaneously. Nonlinear equations are equations in which the highest power of the variable(s) is greater than 1.

Q: How do I solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including the method of substitution, the method of elimination, and the method of graphing. The method of elimination is a common method used to solve systems of linear equations.

Q: What is the intersection point of two lines?

A: The intersection point of two lines is the point where the two lines meet. It is the solution to the system of linear equations.

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

A: To find the intersection point of two lines, you need to solve the system of linear equations. You can use the method of substitution, the method of elimination, or the method of graphing to solve the system.

Q: What is the difference between a dependent system and an independent system?

A: A dependent system is a system of linear equations in which the two equations are equivalent. An independent system is a system of linear equations in which the two equations are not equivalent.

Q: How do I determine if a system of linear equations is dependent or independent?

A: To determine if a system of linear equations is dependent or independent, you need to check if the two equations are equivalent. If the two equations are equivalent, then the system is dependent. If the two equations are not equivalent, then the system is independent.

Q: What is the solution to a dependent system?

A: The solution to a dependent system is all the points on the line that satisfies both equations.

Q: What is the solution to an independent system?

A: The solution to an independent system is a single point that satisfies both equations.

Q: How do I graph a dependent system?

A: To graph a dependent system, you need to graph one of the equations and then draw the line that satisfies both equations.

Q: How do I graph an independent system?

A: To graph an independent system, you need to graph both equations and then find the intersection point of the two lines.

Conclusion

In this article, we have answered some frequently asked questions about graphing and solving systems of linear equations. We have discussed the difference between a system of linear equations and a system of nonlinear equations, and how to solve a system of linear equations using the method of substitution, the method of elimination, and the method of graphing.

We have also discussed the difference between a dependent system and an independent system, and how to determine if a system of linear equations is dependent or independent.

We hope that this article has been helpful in answering your questions about graphing and solving systems of linear equations.

Final Answer

The final answer is:

• A system of linear equations is a set of two or more linear equations that are solved simultaneously.
• To graph a system of linear equations, you need to first graph each equation separately and then find the intersection point of the two lines.
• The solution to a dependent system is all the points on the line that satisfies both equations.
• The solution to an independent system is a single point that satisfies both equations.

References

• [1] "Graphing and Solving Systems of Linear Equations" by Math Open Reference
• [2] "Systems of Linear Equations" by Khan Academy
• [3] "Graphing Systems of Linear Equations" by Purplemath