What Is The Point Of Intersection When The System Of Equations Below Is Graphed?$\[ \begin{cases} -x + Y = 4 \\ 6x + Y = -3 \end{cases} \\]A. \[$(1, -3)\$\]B. \[$(-1, 3)\$\]C. \[$(1, 3)\$\]D. \[$(-1, -3)\$\]

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Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. When a system of equations is graphed, the point of intersection is the point where the two lines or curves meet. In this article, we will explore how to find the point of intersection of a system of linear equations.

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. Each equation in the system is a linear equation, which is an equation in which the highest power of the variable is 1. For example:

{ \begin{cases} -x + y = 4 \\ 6x + y = -3 \end{cases} \}

How to Solve a System of Linear Equations

There are several methods to solve a system of linear equations, including substitution, elimination, and graphing. In this article, we will use the substitution method to solve the system of equations.

Step 1: Write Down the Equations

The first step in solving a system of linear equations is to write down the equations. In this case, we have two equations:

{ \begin{cases} -x + y = 4 \\ 6x + y = -3 \end{cases} \}

Step 2: Solve One Equation for One Variable

The next step is to solve one equation for one variable. Let's solve the first equation for y:

{ -x + y = 4 \}

{ y = x + 4 \}

Step 3: Substitute the Expression into the Other Equation

Now that we have an expression for y, we can substitute it into the other equation:

{ 6x + y = -3 \}

{ 6x + (x + 4) = -3 \}

Step 4: Simplify the Equation

Now that we have substituted the expression for y, we can simplify the equation:

{ 6x + x + 4 = -3 \}

{ 7x + 4 = -3 \}

Step 5: Solve for x

Now that we have simplified the equation, we can solve for x:

{ 7x + 4 = -3 \}

{ 7x = -7 \}

{ x = -1 \}

Step 6: Find the Value of y

Now that we have found the value of x, we can find the value of y by substituting x into one of the original equations:

{ y = x + 4 \}

{ y = -1 + 4 \}

{ y = 3 \}

The Point of Intersection

The point of intersection is the point where the two lines meet. In this case, the point of intersection is (-1, 3).

Conclusion

In this article, we have explored how to find the point of intersection of a system of linear equations. We have used the substitution method to solve the system of equations and have found the point of intersection to be (-1, 3). This is an important concept in mathematics, as it allows us to solve systems of equations and find the values of the variables.

Answer

The correct answer is B. (-1, 3).

Discussion

This problem is a great example of how to use the substitution method to solve a system of linear equations. It is also a great example of how to find the point of intersection of two lines. The point of intersection is an important concept in mathematics, as it allows us to solve systems of equations and find the values of the variables.

Related Topics

  • Solving systems of linear equations using the elimination method
  • Solving systems of linear equations using the graphing method
  • Finding the point of intersection of two lines
  • Solving systems of linear equations with three variables

References

  • [1] "Solving Systems of Linear Equations" by Math Open Reference
  • [2] "Systems of Linear Equations" by Khan Academy
  • [3] "Point of Intersection" by Wolfram MathWorld
    Frequently Asked Questions: 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 to find the values of the variables.

Q: How do I know if a system of linear equations has a solution?

A: A system of linear equations has a solution if the two lines intersect at a single point. If the lines are parallel, the system has no solution.

Q: What is the point of intersection?

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

Q: How do I find the point of intersection?

A: To find the point of intersection, you can use the substitution method, elimination method, or graphing method. The substitution method involves solving one equation for one variable and substituting it into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable. The graphing method involves graphing the two lines and finding the point of intersection.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves solving one equation for one variable and substituting it into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: Can a system of linear equations have more than one solution?

A: No, a system of linear equations can only have one solution. If the lines intersect at more than one point, the system has no solution.

Q: Can a system of linear equations have no solution?

A: Yes, a system of linear equations can have no solution if the lines are parallel.

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, while a system of nonlinear equations is a set of nonlinear equations. Nonlinear equations are equations that are not linear, such as quadratic equations or exponential equations.

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

A: To solve a system of nonlinear equations, you can use numerical methods such as the Newton-Raphson method or the bisection method. These methods involve making an initial guess and then iteratively improving the guess until the solution is found.

Q: What is the importance of solving systems of linear equations?

A: Solving systems of linear equations is important in many fields, including physics, engineering, economics, and computer science. It is used to model real-world problems and to make predictions about the behavior of complex systems.

Q: Can I use a calculator to solve a system of linear equations?

A: Yes, you can use a calculator to solve a system of linear equations. Many calculators have built-in functions for solving systems of linear equations.

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

A: A system of linear equations is a set of linear equations, while a matrix equation is an equation that involves a matrix. A matrix is a rectangular array of numbers.

Q: How do I solve a matrix equation?

A: To solve a matrix equation, you can use methods such as Gaussian elimination or LU decomposition. These methods involve transforming the matrix into a simpler form and then solving for the variables.

Q: What is the importance of solving matrix equations?

A: Solving matrix equations is important in many fields, including physics, engineering, economics, and computer science. It is used to model real-world problems and to make predictions about the behavior of complex systems.

Conclusion

In this article, we have answered some of the most frequently asked questions about systems of linear equations. We have discussed the importance of solving systems of linear equations, the different methods for solving them, and the importance of solving matrix equations. We hope that this article has been helpful in answering your questions and providing you with a better understanding of systems of linear equations.