{ \begin{aligned} y &= 5x - 1 \\ -15x - 3y &= 3 \end{aligned} \}$How Many Solutions Does This Linear System Have?A. One Solution: ${$(0, -1)\$}$B. One Solution: ${$(1, 4)\$}$C. No SolutionD. Infinite Number Of Solutions
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Introduction
Linear systems are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A linear system consists of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will explore the concept of linear systems and provide a step-by-step guide on how to solve them.
What is a Linear System?
A linear system is a set of two or more linear equations that are solved simultaneously to find the values of the variables. Linear equations are equations in which the highest power of the variable is 1. For example, the equation 2x + 3y = 5 is a linear equation, while the equation x^2 + 2y = 3 is not a linear equation.
Types of Linear Systems
There are three types of linear systems:
- Consistent System: A consistent system is a linear system that has at least one solution. In other words, there is a set of values for the variables that satisfies all the equations in the system.
- Inconsistent System: An inconsistent system is a linear system that has no solution. In other words, there is no set of values for the variables that satisfies all the equations in the system.
- Dependent System: A dependent system is a linear system that has an infinite number of 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
There are several methods to solve linear systems, including:
- Substitution Method: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
- Elimination Method: The elimination method involves adding or subtracting the equations to eliminate one variable.
- Graphical Method: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection.
Example: Solving a Linear System
Let's consider the following linear system:
To solve this system, we can use the substitution method. First, we can solve the first equation for y:
Next, we can substitute this expression into the second equation:
Expanding and simplifying the equation, we get:
Combine like terms:
Subtract 3 from both sides:
Divide both sides by -30:
Now that we have found 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:
Substitute x = 0:
Simplify:
Therefore, the solution to the linear system is (0, -1).
Conclusion
In this article, we have explored the concept of linear systems and provided a step-by-step guide on how to solve them. We have also discussed the different types of linear systems and the methods used to solve them. By following the steps outlined in this article, you should be able to solve linear systems with ease.
Final Answer
The final answer is:
- A. One solution: (0, -1)
This is the correct answer because the linear system has a consistent solution, and the values of x and y that satisfy the system are x = 0 and y = -1.
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Introduction
Linear systems are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will answer some of the most frequently asked questions about linear systems.
Q1: What is a Linear System?
A linear system is a set of two or more linear equations that are solved simultaneously to find the values of the variables. Linear equations are equations in which the highest power of the variable is 1.
Q2: What are the Types of Linear Systems?
There are three types of linear systems:
- Consistent System: A consistent system is a linear system that has at least one solution. In other words, there is a set of values for the variables that satisfies all the equations in the system.
- Inconsistent System: An inconsistent system is a linear system that has no solution. In other words, there is no set of values for the variables that satisfies all the equations in the system.
- Dependent System: A dependent system is a linear system that has an infinite number of solutions. In other words, there are an infinite number of sets of values for the variables that satisfy all the equations in the system.
Q3: How Do I Solve a Linear System?
There are several methods to solve linear systems, including:
- Substitution Method: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.
- Elimination Method: The elimination method involves adding or subtracting the equations to eliminate one variable.
- Graphical Method: The graphical method involves graphing the equations on a coordinate plane and finding the point of intersection.
Q4: What is the Difference Between a Consistent and Inconsistent System?
A consistent system is a linear system that has at least one solution, while an inconsistent system is a linear system that has no solution.
Q5: How Do I Determine if a Linear System is Consistent or Inconsistent?
To determine if a linear system is consistent or inconsistent, you can use the following methods:
- Substitution Method: Substitute the expression for one variable into the other equation and solve for the other variable.
- Elimination Method: Add or subtract the equations to eliminate one variable and solve for the other variable.
- Graphical Method: Graph the equations on a coordinate plane and find the point of intersection.
Q6: What is the Difference Between a Dependent and Independent System?
A dependent system is a linear system that has an infinite number of solutions, while an independent system is a linear system that has a unique solution.
Q7: How Do I Determine if a Linear System is Dependent or Independent?
To determine if a linear system is dependent or independent, you can use the following methods:
- Substitution Method: Substitute the expression for one variable into the other equation and solve for the other variable.
- Elimination Method: Add or subtract the equations to eliminate one variable and solve for the other variable.
- Graphical Method: Graph the equations on a coordinate plane and find the point of intersection.
Q8: Can a Linear System Have More Than One Solution?
Yes, a linear system can have more than one solution. In fact, a linear system can have an infinite number of solutions if the equations are dependent.
Q9: Can a Linear System Have No Solution?
Yes, a linear system can have no solution. In fact, a linear system can have no solution if the equations are inconsistent.
Q10: How Do I Graph a Linear System?
To graph a linear system, you can use the following steps:
- Step 1: Graph the first equation on a coordinate plane.
- Step 2: Graph the second equation on the same coordinate plane.
- Step 3: Find the point of intersection between the two graphs.
Conclusion
In this article, we have answered some of the most frequently asked questions about linear systems. We have discussed the different types of linear systems, the methods used to solve them, and how to determine if a linear system is consistent, inconsistent, dependent, or independent. By following the steps outlined in this article, you should be able to solve linear systems with ease.
Final Answer
The final answer is:
- A. One solution: (0, -1)
This is the correct answer because the linear system has a consistent solution, and the values of x and y that satisfy the system are x = 0 and y = -1.