State The Number Of Solutions For The Linear System Of Equations:$\[ \begin{align*} 12y - 168x &= -72 \\ 12y - 168x &= 132 \end{align*} \\]A. No Solutions B. One Solution C. Infinite Solutions

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Introduction

Linear systems of equations are a fundamental concept in mathematics, and understanding the number of solutions they have is crucial for solving them. In this article, we will explore the concept of linear systems of equations and determine the number of solutions for a given system.

What are Linear Systems of Equations?

A linear system of equations is a set of two or more linear equations that involve the same variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables. The system of equations is said to be linear if all the equations are linear.

Types of Linear Systems of Equations

There are three types of linear systems of equations:

  • Consistent System: A consistent system of equations has at least one solution. In other words, there exists at least one set of values for the variables that satisfies all the equations in the system.
  • Inconsistent System: An inconsistent system of equations has no solution. In other words, there does not exist any set of values for the variables that satisfies all the equations in the system.
  • Dependent System: A dependent system of equations has infinitely many solutions. In other words, there are an infinite number of sets of values for the variables that satisfy all the equations in the system.

Solving Linear Systems of Equations

To solve a linear system of equations, we can use various methods such as substitution, elimination, and graphing. The method we choose depends on the type of system we are dealing with.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when one equation is easily solvable for one variable.

Elimination Method

The elimination method involves adding or subtracting the equations in the system to eliminate one variable. This method is useful when the coefficients of one variable are the same in both equations.

Graphing Method

The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when we want to visualize the solution.

Determining the Number of Solutions

To determine the number of solutions for a linear system of equations, we can use the following criteria:

  • If the equations are identical, then the system has infinitely many solutions.
  • If the equations are parallel, then the system has no solution.
  • If the equations intersect, then the system has exactly one solution.

Example

Let's consider the following system of equations:

{ \begin{align*} 12y - 168x &= -72 \\ 12y - 168x &= 132 \end{align*} \}

To determine the number of solutions for this system, we can first simplify the equations by dividing both sides of each equation by 12.

{ \begin{align*} y - 14x &= -6 \\ y - 14x &= 11 \end{align*} \}

Now, we can see that the two equations are identical. Therefore, the system has infinitely many solutions.

Conclusion

In conclusion, determining the number of solutions for a linear system of equations is crucial for solving them. By understanding the types of linear systems of equations and using various methods such as substitution, elimination, and graphing, we can determine the number of solutions for a given system. In this article, we have explored the concept of linear systems of equations and determined the number of solutions for a given system.

References

Introduction

Linear systems of equations are a fundamental concept in mathematics, and understanding them is crucial for solving them. In this article, we will answer some frequently asked questions about linear systems of equations.

Q: What is a linear system of equations?

A: A linear system of equations is a set of two or more linear equations that involve the same variables. Each equation is in the form of ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: What are the types of linear systems of equations?

A: There are three types of linear systems of equations:

  • Consistent System: A consistent system of equations has at least one solution. In other words, there exists at least one set of values for the variables that satisfies all the equations in the system.
  • Inconsistent System: An inconsistent system of equations has no solution. In other words, there does not exist any set of values for the variables that satisfies all the equations in the system.
  • Dependent System: A dependent system of equations has infinitely many solutions. In other words, there are an infinite number of sets of values for the variables that satisfy all the equations in the system.

Q: How do I determine the number of solutions for a linear system of equations?

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

  • If the equations are identical, then the system has infinitely many solutions.
  • If the equations are parallel, then the system has no solution.
  • If the equations intersect, then the system has exactly one solution.

Q: What are the methods for solving linear systems of equations?

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

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

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

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

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

A: A dependent system of equations has infinitely many solutions, while an independent system of equations has exactly one solution.

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

A: No, a linear system of equations can have at most one solution.

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

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

Q: Can a linear system of equations have infinitely many solutions?

A: Yes, a linear system of equations can have infinitely many solutions if the equations are dependent.

Conclusion

In conclusion, linear systems of equations are a fundamental concept in mathematics, and understanding them is crucial for solving them. By answering some frequently asked questions about linear systems of equations, we have provided a better understanding of this concept.

References