Solve This System Of Equations:${ \begin{cases} 2x - 3y = 0 \ -3x + 3y = -3 \end{cases} }$Choose One Of The Following Options:A. One Or More Solutions B. No Solution C. Infinite Number Of Solutions

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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 involves 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{cases} 2x - 3y = 0 \\ -3x + 3y = -3 \end{cases} \}

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 first equation for x:

2x - 3y = 0

2x = 3y

x = (3/2)y

Now, substitute this expression for x into the second equation:

-3x + 3y = -3

-3((3/2)y) + 3y = -3

-9y/2 + 3y = -3

-9y/2 + 6y/2 = -3

-3y/2 = -3

-3y = -6

y = 2

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

2x - 3y = 0

2x - 3(2) = 0

2x - 6 = 0

2x = 6

x = 3

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 in a way that eliminates one of the variables.

Let's start by multiplying the first equation by 3 and the second equation by 2:

6x - 9y = 0

-6x + 6y = -6

Now, add the two equations together:

-9y + 6y = -6

-3y = -6

y = 2

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

2x - 3y = 0

2x - 3(2) = 0

2x - 6 = 0

2x = 6

x = 3

Conclusion

In this article, we have solved a system of two linear equations with two variables using two different methods: the substitution method and the elimination method. Both methods have led to the same solution: x = 3 and y = 2.

Final Answer

The final answer is:

  • One or more solutions: The system of equations has a unique solution: x = 3 and y = 2.

Discussion

This system of equations is a classic example of a linear system with a unique solution. The two equations are linearly independent, meaning that they are not multiples of each other. As a result, the system has a unique solution, which can be found using either the substitution method or the elimination method.

Related Topics

  • Linear Equations: Linear equations are equations in which the highest power of the variable(s) is 1.
  • Systems of Linear Equations: A system of linear equations is a set of two or more linear equations that involve the same set of variables.
  • Substitution Method: The substitution method is a technique used to solve systems of linear equations by solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination Method: The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations in a way that eliminates one of the variables.

Glossary

  • Linear Equation: A linear equation is an equation in which the highest power of the variable(s) is 1.
  • System of Linear Equations: A system of linear equations is a set of two or more linear equations that involve the same set of variables.
  • Substitution Method: The substitution method is a technique used to solve systems of linear equations by solving one equation for one variable and then substituting that expression into the other equation.
  • Elimination Method: The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations in a way that eliminates one of the variables.

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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 in the system is a linear equation, meaning that the highest power of the variable(s) is 1.

Q: How do I know if a system of linear equations has a unique solution, no solution, or infinite solutions?

A: To determine the number of solutions to a system of linear equations, you can use the following criteria:

  • Unique solution: If the two equations are linearly independent (i.e., they are not multiples of each other), then the system has a unique solution.
  • No solution: If the two equations are linearly dependent (i.e., one equation is a multiple of the other), then the system has no solution.
  • Infinite solutions: If the two equations are identical (i.e., they are the same equation), then the system has infinite solutions.

Q: What is the substitution method?

A: The substitution method is a technique used to solve systems of linear equations by solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting the equations in a way that eliminates one of the variables.

Q: How do I choose between the substitution method and the elimination method?

A: The choice between the substitution method and the elimination method depends on the specific system of linear equations you are working with. If one equation is already solved for one variable, then the substitution method may be easier to use. If the coefficients of the variables are easy to work with, then the elimination method may be easier to use.

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 equations are linearly independent: If the equations are linearly dependent, then the system may have no solution or infinite solutions.
  • Not using the correct method: Using the wrong method can lead to incorrect solutions.
  • Not checking for extraneous solutions: If the system has infinite solutions, then any solution that satisfies both equations is an extraneous solution.

Q: How do I check if a solution is extraneous?

A: To check if a solution is extraneous, you can substitute the solution back into both equations and check if it satisfies both equations.

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 and engineering: Solving systems of linear equations is used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
  • Economics: Solving systems of linear equations is used to model economic systems and make predictions about economic trends.
  • Computer science: Solving systems of linear equations is used in computer graphics and game development to create realistic simulations.

Q: How do I know if a system of linear equations is consistent or inconsistent?

A: A system of linear equations is consistent if it has at least one solution. A system of linear equations is inconsistent if it has no solution.

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

A: A consistent system of linear equations has at least one solution, while an inconsistent system of linear equations has no solution.

Q: How do I determine if a system of linear equations is consistent or inconsistent?

A: To determine if a system of linear equations is consistent or inconsistent, you can use the following criteria:

  • Consistent: If the system has at least one solution, then it is consistent.
  • Inconsistent: If the system has no solution, then it is inconsistent.

Q: What are some common types of systems of linear equations?

A: Some common types of systems of linear equations include:

  • Homogeneous systems: A homogeneous system of linear equations is a system in which all the constant terms are zero.
  • Nonhomogeneous systems: A nonhomogeneous system of linear equations is a system in which not all the constant terms are zero.
  • Linearly independent systems: A linearly independent system of linear equations is a system in which the equations are not multiples of each other.
  • Linearly dependent systems: A linearly dependent system of linear equations is a system in which the equations are multiples of each other.