Why the Solution is Not Unique

Oct 13, 2025 by windailys.com 65 views

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will focus on solving a system of two linear equations with two variables. We will use the given system of equations to demonstrate the steps involved in solving a system of linear equations.

The System of Equations

The given system of equations is:

\begin{array}{l} 3x + 8y = 9 \\ 4x - 8y = 12 \end{array}

Step 1: Write Down the System of Equations

The first step in solving a system of linear equations is to write down the system of equations. In this case, we have two linear equations with two variables, x and y.

Step 2: Add the Two Equations

To eliminate one of the variables, we can add the two equations together. This will eliminate the variable y.

\begin{array}{l} (3x + 8y) + (4x - 8y) = 9 + 12 \\ 7x = 21 \end{array}

Step 3: Solve for x

Now that we have eliminated the variable y, we can solve for x. To do this, we can divide both sides of the equation by 7.

\begin{array}{l} 7x = 21 \\ x = \frac{21}{7} \\ x = 3 \end{array}

Step 4: Substitute x into One of the Original Equations

Now that we have found the value of x, we can substitute it into one of the original equations to solve for y. We will use the first equation.

\begin{array}{l} 3x + 8y = 9 \\ 3(3) + 8y = 9 \\ 9 + 8y = 9 \\ 8y = 0 \\ y = 0 \end{array}

Conclusion

In this article, we have demonstrated the steps involved in solving a system of linear equations. We used the given system of equations to show how to add the two equations together to eliminate one of the variables, solve for the other variable, and substitute the value of one variable into one of the original equations to solve for the other variable. The solution to the system of equations is x = 3 and y = 0.

The Final Answer

The final answer is x = 3 and y = 0.

Why the Solution is Not Unique

In this case, the solution to the system of equations is unique, meaning that there is only one solution. However, in some cases, the solution to a system of linear equations may not be unique. This can happen when the two equations are identical or when the system of equations has no solution.

When the System of Equations Has No Solution

A system of linear equations may have no solution when the two equations are inconsistent. This can happen when the two equations represent parallel lines that never intersect.

When the System of Equations Has an Infinite Number of Solutions

A system of linear equations may have an infinite number of solutions when the two equations are identical. This can happen when the two equations represent the same line.

Conclusion

In conclusion, solving a system of linear equations involves adding the two equations together to eliminate one of the variables, solving for the other variable, and substituting the value of one variable into one of the original equations to solve for the other variable. The solution to the system of equations may be unique, or it may not be unique, depending on the nature of the system of equations.

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra: A Modern Introduction" by David Poole

Additional Resources

Khan Academy: Linear Equations
MIT OpenCourseWare: Linear Algebra
Wolfram MathWorld: Linear Equations
Solving a System of Linear Equations: Q&A =====================================

Introduction

In our previous article, we discussed how to solve a system of linear equations. In this article, we will answer some frequently asked questions about solving a system of linear equations.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

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

To determine the nature of the solution, you can use the following methods:

If the two equations are identical, the system has an infinite number of solutions.
If the two equations are inconsistent, the system has no solution.
If the two equations are consistent and not identical, the system has a unique solution.

Q: How do I add two equations together to eliminate one of the variables?

To add two equations together, you can use the following steps:

Multiply both equations by the necessary multiples to make the coefficients of the variable you want to eliminate the same.
Add the two equations together.
Simplify the resulting equation.

Q: How do I solve for the variable after eliminating one of the variables?

To solve for the variable, you can use the following steps:

Divide both sides of the equation by the coefficient of the variable.
Simplify the resulting equation.

Q: Can I use substitution or elimination to solve a system of linear equations?

Yes, you can use either substitution or elimination to solve a system of linear equations. The choice of method depends on the nature of the system and the variables involved.

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

A system of linear equations is consistent if it has a solution. A system of linear equations is inconsistent if it has no solution.

Q: Can a system of linear equations have an infinite number of solutions?

Yes, a system of linear equations can have an infinite number of solutions if the two equations are identical.

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

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

Graph each equation separately.
Find the point of intersection of the two lines.
The point of intersection represents the solution to the system.

Q: Can I use technology to solve a system of linear equations?

Yes, you can use technology such as calculators or computer software to solve a system of linear equations.

Conclusion

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra: A Modern Introduction" by David Poole

Additional Resources

Khan Academy: Linear Equations
MIT OpenCourseWare: Linear Algebra
Wolfram MathWorld: 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 are solved simultaneously to find the values of the variables.
Q: How do I know if a system of linear equations has a unique solution, no solution, or an infinite number of solutions? A: To determine the nature of the solution, you can use the following methods: if the two equations are identical, the system has an infinite number of solutions; if the two equations are inconsistent, the system has no solution; if the two equations are consistent and not identical, the system has a unique solution.
Q: How do I add two equations together to eliminate one of the variables? A: To add two equations together, you can use the following steps: multiply both equations by the necessary multiples to make the coefficients of the variable you want to eliminate the same; add the two equations together; simplify the resulting equation.
Q: How do I solve for the variable after eliminating one of the variables? A: To solve for the variable, you can use the following steps: divide both sides of the equation by the coefficient of the variable; simplify the resulting equation.