Solve The System Of Equations:${ \begin{array}{l} -3x + 3y = 0 \ -x + Y = 3 \end{array} }$

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

The System of Equations

The system of equations we will be solving is:

$$\begin{array}{l} -3x + 3y = 0 \\ -x + y = 3 \end{array}$$

Method 1: Substitution Method

One way to solve this system of equations is by using the substitution method. This method involves solving one of the equations for one variable and then substituting that expression into the other equation.

Let's start by solving the second equation for y:

$$y = x + 3$$

Now, substitute this expression for y into the first equation:

$$-3x + 3(x + 3) = 0$$

Expand and simplify the equation:

$$-3x + 3x + 9 = 0$$

Combine like terms:

$$9 = 0$$

This is a contradiction, which means that the system of equations has no solution.

Method 2: Elimination Method

Another way to solve this system of equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one of the variables.

Let's start by multiplying the first equation by 1 and the second equation by 3:

$$-3x + 3y = 0$$

$$-3x + 3y = 9$$

Now, add the two equations to eliminate the x variable:

$$-6x + 6y = 9$$

This is a contradiction, which means that the system of equations has no solution.

Method 3: Graphical Method

Another way to solve this system of equations is by using the graphical method. This method involves graphing the two equations on a coordinate plane and finding the point of intersection.

Let's start by graphing the two equations:

The first equation is a line with a slope of 1 and a y-intercept of 0:

$$y = \frac{1}{3}x$$

The second equation is a line with a slope of 1 and a y-intercept of 3:

$$y = x + 3$$

The two lines intersect at the point (0, 0), but this is not a solution to the system of equations.

Conclusion

In this article, we have seen three different methods for solving a system of linear equations. The substitution method, elimination method, and graphical method all led to the same conclusion: the system of equations has no solution.

Why the System of Equations Has No Solution

The system of equations has no solution because the two equations are inconsistent. The first equation is a line with a slope of 1 and a y-intercept of 0, while the second equation is a line with a slope of 1 and a y-intercept of 3. These two lines are parallel and never intersect, which means that there is no point that satisfies both equations.

Real-World Applications

Solving systems of linear equations has many real-world applications. For example, in physics, the motion of an object can be described by a system of linear equations. In economics, the supply and demand of a product can be described by a system of linear equations. In engineering, the design of a bridge or a building can be described by a system of linear equations.

Final Thoughts

In conclusion, solving a system of linear equations is an important skill in mathematics and has many real-world applications. The substitution method, elimination method, and graphical method are all useful tools for solving systems of linear equations. However, in this case, the system of equations has no solution because the two equations are inconsistent.

Additional Resources

• Khan Academy: Systems of Linear Equations
• MIT OpenCourseWare: Linear Algebra
• Wolfram Alpha: Systems of Linear Equations

Frequently Asked Questions

• Q: What is a system of linear equations?
• A: A system of linear equations is a set of two or more linear equations that involve the same set of variables.
• Q: How do I solve a system of linear equations?
• A: There are several methods for solving a system of linear equations, including the substitution method, elimination method, and graphical method.
• Q: Why do some systems of linear equations have no solution?
• A: A system of linear equations has no solution if the two equations are inconsistent, meaning that they do not intersect at any point.
  Frequently Asked Questions: 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 involve the same set of variables. Each equation is a linear equation, meaning that 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 solve a system of linear equations?

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

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

Q: Why do some systems of linear equations have no solution?

A: A system of linear equations has no solution if the two equations are inconsistent, meaning that they do not intersect at any point. This can happen if the two equations are parallel lines that never intersect.

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, meaning that each equation can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables. A system of nonlinear equations, on the other hand, is a set of nonlinear equations, meaning that each equation cannot be written in the form ax + by = c.

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

A: To determine if a system of linear equations has a unique solution, no solution, or infinitely many solutions, you can use the following criteria:

• Unique Solution: If the two equations are not parallel and intersect at a single point, then the system has a unique solution.
• No Solution: If the two equations are parallel and never intersect, then the system has no solution.
• Infinitely Many Solutions: If the two equations are the same, then the system has infinitely many solutions.

Q: What are some real-world applications of solving systems of linear equations?

A: Solving systems of linear equations has many real-world applications, including:

• Physics: The motion of an object can be described by a system of linear equations.
• Economics: The supply and demand of a product can be described by a system of linear equations.
• Engineering: The design of a bridge or a building can be described by a system of linear equations.

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

A: To graph a system of linear equations, you can use the following steps:

1. Graph the first equation: Graph the first equation on a coordinate plane.
2. Graph the second equation: Graph the second equation on the same coordinate plane.
3. Find the point of intersection: Find the point where the two lines intersect.

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 for consistency: Make sure that the two equations are consistent before trying to solve them.
• Not using the correct method: Choose the correct method for solving the system, such as substitution or elimination.
• Not checking for infinitely many solutions: Make sure that the system does not have infinitely many solutions before trying to find a unique solution.

Additional Resources

• Khan Academy: Systems of Linear Equations
• MIT OpenCourseWare: Linear Algebra
• Wolfram Alpha: Systems of Linear Equations

Practice Problems

• Solve the following system of linear equations: x + 2y = 4, 3x - 2y = -2.
• Solve the following system of linear equations: 2x + 3y = 5, x - 2y = -3.
• Solve the following system of linear equations: x + y = 2, 2x + 3y = 7.

Conclusion

Solving systems of linear equations is an important skill in mathematics and has many real-world applications. By understanding the different methods for solving systems of linear equations, you can apply this knowledge to a variety of problems in physics, economics, and engineering. Remember to check for consistency, use the correct method, and check for infinitely many solutions to ensure that you are solving the system correctly.