Is { (1, -1)$}$ A Solution To This System Of Equations?${ \begin{array}{l} y = -6x + 5 \ y = X - 2 \end{array} }$A. Yes B. No

by ADMIN 128 views

Introduction

When dealing with systems of linear equations, it's essential to determine whether a given point satisfies both equations. In this case, we're presented with a system of two linear equations and asked to verify if the point {(1, -1)$}$ is a solution to the system. To do this, we'll substitute the coordinates of the given point into both equations and check if the resulting statements are true.

Understanding the System of Equations

The given system of equations consists of two linear equations in the form of:

{ \begin{array}{l} y = -6x + 5 \\ y = x - 2 \end{array} \}

These equations represent lines on a coordinate plane, and we're interested in finding out if the point {(1, -1)$}$ lies on both lines.

Substituting the Point into the First Equation

To verify if the point {(1, -1)$}$ satisfies the first equation, we'll substitute the x-coordinate (1) and the y-coordinate (-1) into the equation y=−6x+5y = -6x + 5. This gives us:

{ -1 = -6(1) + 5 \}

Simplifying the equation, we get:

{ -1 = -6 + 5 \}

{ -1 = -1 \}

As we can see, the statement is true, and the point {(1, -1)$}$ satisfies the first equation.

Substituting the Point into the Second Equation

Next, we'll substitute the x-coordinate (1) and the y-coordinate (-1) into the second equation y=x−2y = x - 2. This gives us:

{ -1 = 1 - 2 \}

Simplifying the equation, we get:

{ -1 = -1 \}

Again, the statement is true, and the point {(1, -1)$}$ satisfies the second equation.

Conclusion

Since the point {(1, -1)$}$ satisfies both equations in the system, we can conclude that it is indeed a solution to the system of equations.

Final Answer

The final answer is: A. Yes

Discussion

In this discussion, we've demonstrated how to verify if a given point satisfies a system of linear equations. By substituting the coordinates of the point into both equations, we can determine whether the point lies on both lines. This is an essential skill in mathematics, particularly in algebra and geometry.

Related Topics

  • Systems of Linear Equations
  • Linear Equations
  • Coordinate Geometry
  • Algebra

Further Reading

For more information on systems of linear equations and coordinate geometry, we recommend the following resources:

  • Khan Academy: Systems of Linear Equations
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram MathWorld: Coordinate Geometry

Exercises

  1. Verify if the point {(2, 3)$}$ is a solution to the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = x + 2 \end{array} \}

  1. Find the solution to the system of equations:

{ \begin{array}{l} y = 3x - 2 \\ y = 2x + 1 \end{array} \}

Solutions

  1. To verify if the point {(2, 3)$}$ is a solution to the system of equations, we'll substitute the x-coordinate (2) and the y-coordinate (3) into both equations. This gives us:

{ 3 = 2(2) - 1 \}

{ 3 = 4 - 1 \}

{ 3 = 3 \}

As we can see, the statement is true, and the point {(2, 3)$}$ satisfies the first equation.

Next, we'll substitute the x-coordinate (2) and the y-coordinate (3) into the second equation y=x+2y = x + 2. This gives us:

{ 3 = 2 + 2 \}

{ 3 = 4 \}

Unfortunately, the statement is false, and the point {(2, 3)$}$ does not satisfy the second equation.

  1. To find the solution to the system of equations, we'll first equate the two expressions for y:

{ 3x - 2 = 2x + 1 \}

Next, we'll solve for x by isolating the variable:

{ 3x - 2x = 1 + 2 \}

{ x = 3 \}

Now that we have the value of x, we can substitute it into one of the original equations to find the value of y. Let's use the first equation:

{ y = 3x - 2 \}

{ y = 3(3) - 2 \}

{ y = 9 - 2 \}

{ y = 7 \}

Therefore, the solution to the system of equations is {(3, 7)$}$.

Introduction

Systems of linear equations are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra, geometry, and other areas of mathematics. In this article, we'll answer some of the most frequently asked questions about systems of linear equations, providing you with a deeper understanding of this important topic.

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. Each equation in the system is a linear equation, which means it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

A: To determine if a system of linear equations has a solution, you need to check if the two equations are consistent with each other. If the equations are consistent, then the system has a solution. If the equations are inconsistent, then the system does not have a solution.

Q: What is the difference between a consistent and an inconsistent system of linear equations?

A: A consistent system of linear equations is one that has a solution, while an inconsistent system of linear equations is one that does not have a solution. For example, the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = x + 2 \end{array} \}

is consistent, while the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = x + 3 \end{array} \}

is inconsistent.

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

A: There are several methods for solving a system of linear equations, including:

  • Substitution method: This involves substituting the expression for one variable from one equation into the other equation.
  • Elimination method: This involves adding or subtracting the two equations to eliminate one of the variables.
  • Graphical method: This involves graphing the two equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method involves substituting the expression for one variable from one equation into the other equation. For example, if we have the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = x + 2 \end{array} \}

we can substitute the expression for y from the first equation into the second equation:

{ 2x - 1 = x + 2 \}

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the two equations to eliminate one of the variables. For example, if we have the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = x + 2 \end{array} \}

we can add the two equations to eliminate the variable y:

{ 2x - 1 + x + 2 = 0 \}

Q: What is the graphical method?

A: The graphical method involves graphing the two equations on a coordinate plane and finding the point of intersection. For example, if we have the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = x + 2 \end{array} \}

we can graph the two equations on a coordinate plane and find the point of intersection.

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

A: Some common mistakes to avoid when solving systems of linear equations include:

  • Not checking if the system has a solution before solving it.
  • Not using the correct method for solving the system (e.g. substitution, elimination, or graphical).
  • Not checking if the solution satisfies both equations.
  • Not simplifying the equations before solving them.

Q: How do I know if a system of linear equations has infinitely many solutions?

A: A system of linear equations has infinitely many solutions if the two equations are equivalent, meaning that one equation is a multiple of the other. For example, the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = 2(2x - 1) \end{array} \}

has infinitely many solutions.

Q: How do I know if a system of linear equations has no solution?

A: A system of linear equations has no solution if the two equations are inconsistent, meaning that they cannot be true at the same time. For example, the system of equations:

{ \begin{array}{l} y = 2x - 1 \\ y = x + 3 \end{array} \}

has no solution.

Conclusion

Solving systems of linear equations is an essential skill in mathematics, and understanding how to solve them is crucial for success in algebra, geometry, and other areas of mathematics. By following the steps outlined in this article, you can develop a deeper understanding of systems of linear equations and improve your problem-solving skills.