Determine If The Following System Of Equations Has No Solutions, Infinitely Many Solutions, Or Exactly One Solution.${ \begin{aligned} -x + 3y &= -1 \ 4x - 12y &= 3 \end{aligned} }$

Mar 8, 2025 by ADMIN 183 views

**Determine the Nature of Solutions for a System of Linear Equations**

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 simultaneously. In this article, we will determine the nature of solutions for a given system of linear equations.

The System of Equations

The given system of equations is:

{ \begin{aligned} -x + 3y &= -1 \\ 4x - 12y &= 3 \end{aligned} \}

Method 1: Graphical Method

To determine the nature of solutions, we can use the graphical method. We will graph the two equations on the same coordinate plane and observe the intersection points.

Step 1: Graph the First Equation

The first equation is $-x + 3y = -1$ . We can rewrite it in slope-intercept form as $y = \frac{1}{3}x + \frac{1}{3}$ . The slope of this line is $\frac{1}{3}$ , and the y-intercept is $\frac{1}{3}$ .

Step 2: Graph the Second Equation

The second equation is $4x - 12y = 3$ . We can rewrite it in slope-intercept form as $y = \frac{1}{3}x - \frac{1}{4}$ . The slope of this line is $\frac{1}{3}$ , and the y-intercept is $-\frac{1}{4}$ .

Step 3: Observe the Intersection Points

By graphing the two equations, we can observe that they intersect at a single point. This means that the system of equations has exactly one solution.

Method 2: Algebraic Method

We can also use the algebraic method to determine the nature of solutions. We will use the method of substitution or elimination to solve the system of equations.

Step 1: Multiply the First Equation by 4

We will multiply the first equation by 4 to make the coefficients of $x$ in both equations the same.

{ \begin{aligned} -4x + 12y &= -4 \\ 4x - 12y &= 3 \end{aligned} \}

Step 2: Add the Two Equations

We will add the two equations to eliminate the variable $x$ .

{ \begin{aligned} 0 &= -1 \end{aligned} \}

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

Conclusion

In this article, we determined the nature of solutions for a given system of linear equations using the graphical and algebraic methods. We found that the system of equations has exactly one solution using the graphical method and no solutions using the algebraic method.

Nature of Solutions

The nature of solutions for a system of linear equations can be classified into three categories:

No solutions: The system of equations has no solutions if the lines are parallel and do not intersect.
Infinitely many solutions: The system of equations has infinitely many solutions if the lines coincide.
Exactly one solution: The system of equations has exactly one solution if the lines intersect at a single point.

Example

Consider the system of equations:

{ \begin{aligned} x + y &= 2 \\ x - y &= 1 \end{aligned} \}

Using the algebraic method, we can solve the system of equations by adding the two equations to eliminate the variable $y$ .

{ \begin{aligned} 2x &= 3 \\ x &= \frac{3}{2} \end{aligned} \}

Substituting the value of $x$ into one of the original equations, we can solve for $y$ .

{ \begin{aligned} \frac{3}{2} + y &= 2 \\ y &= \frac{1}{2} \end{aligned} \}

Therefore, the system of equations has exactly one solution, which is $x = \frac{3}{2}$ and $y = \frac{1}{2}$ .

Tips and Tricks

Here are some tips and tricks to help you determine the nature of solutions for a system of linear equations:

Check if the lines are parallel: If the lines are parallel, the system of equations has no solutions.
Check if the lines coincide: If the lines coincide, the system of equations has infinitely many solutions.
Check if the lines intersect: If the lines intersect, the system of equations has exactly one solution.
Use the algebraic method: The algebraic method can be used to solve the system of equations by adding or subtracting the equations to eliminate the variables.
Use the graphical method: The graphical method can be used to visualize the lines and determine the nature of solutions.

Conclusion

Q: What is the nature of solutions for a system of linear equations?

A: The nature of solutions for a system of linear equations can be classified into three categories:

No solutions: The system of equations has no solutions if the lines are parallel and do not intersect.
Infinitely many solutions: The system of equations has infinitely many solutions if the lines coincide.
Exactly one solution: The system of equations has exactly one solution if the lines intersect at a single point.

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

A: You can use the graphical and algebraic methods to determine the nature of solutions for a system of linear equations.

Graphical method: Graph the two equations on the same coordinate plane and observe the intersection points.
Algebraic method: Use the method of substitution or elimination to solve the system of equations.

Q: What is the graphical method?

A: The graphical method involves graphing the two equations on the same coordinate plane and observing the intersection points.

Step 1: Graph the first equation: Graph the first equation on the coordinate plane.
Step 2: Graph the second equation: Graph the second equation on the same coordinate plane.
Step 3: Observe the intersection points: Observe the intersection points of the two lines.

Q: What is the algebraic method?

A: The algebraic method involves using the method of substitution or elimination to solve the system of equations.

Step 1: Multiply the first equation by 4: Multiply the first equation by 4 to make the coefficients of x in both equations the same.
Step 2: Add the two equations: Add the two equations to eliminate the variable x.
Step 3: Solve for y: Solve for y by substituting the value of x into one of the original equations.

Q: What are some tips and tricks for determining the nature of solutions for a system of linear equations?

A: Here are some tips and tricks to help you determine the nature of solutions for a system of linear equations:

Check if the lines are parallel: If the lines are parallel, the system of equations has no solutions.
Check if the lines coincide: If the lines coincide, the system of equations has infinitely many solutions.
Check if the lines intersect: If the lines intersect, the system of equations has exactly one solution.
Use the algebraic method: The algebraic method can be used to solve the system of equations by adding or subtracting the equations to eliminate the variables.
Use the graphical method: The graphical method can be used to visualize the lines and determine the nature of solutions.

Q: What are some common mistakes to avoid when determining the nature of solutions for a system of linear equations?

A: Here are some common mistakes to avoid when determining the nature of solutions for a system of linear equations:

Not checking if the lines are parallel: If the lines are parallel, the system of equations has no solutions.
Not checking if the lines coincide: If the lines coincide, the system of equations has infinitely many solutions.
Not checking if the lines intersect: If the lines intersect, the system of equations has exactly one solution.
Not using the algebraic method: The algebraic method can be used to solve the system of equations by adding or subtracting the equations to eliminate the variables.
Not using the graphical method: The graphical method can be used to visualize the lines and determine the nature of solutions.

Q: How do I apply the concepts learned in this article to real-world problems?

A: The concepts learned in this article can be applied to real-world problems in various fields such as engineering, economics, and computer science.

Linear programming: The concepts learned in this article can be applied to linear programming problems, which involve finding the optimal solution to a system of linear equations.
Optimization problems: The concepts learned in this article can be applied to optimization problems, which involve finding the optimal solution to a system of linear equations.
Data analysis: The concepts learned in this article can be applied to data analysis problems, which involve analyzing data and making predictions based on the data.

Conclusion

In conclusion, determining the nature of solutions for a system of linear equations is an important concept in mathematics. By using the graphical and algebraic methods, we can determine whether the system of equations has no solutions, infinitely many solutions, or exactly one solution. We can also use the tips and tricks provided to help us determine the nature of solutions.