Solve one of the equations for one of the variables

Oct 13, 2025 by miodeya.com 82 views

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system. In this article, we will explore two systems of linear equations and determine how many solutions each system has.

System I

The first system consists of two equations:

${ x = y - 3 }$

${ 2x - 2y = -6 }$

To determine the number of solutions, we can start by solving one of the equations for one of the variables. Let's solve the first equation for x:

${ x = y - 3 }$

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

${ 2(y - 3) - 2y = -6 }$

Expand and simplify the equation:

${ 2y - 6 - 2y = -6 }$

${ -6 = -6 }$

This equation is true for all values of y. Therefore, the first equation is an identity, and the second equation is a contradiction. This means that the system has no solution.

System II

The second system consists of two equations:

${ \begin{array}{l} 5x + 2y = -7 \\ 15x + 6y = 21 \end{array} }$

To determine the number of solutions, we can start by solving one of the equations for one of the variables. Let's solve the first equation for x:

${ 5x + 2y = -7 }$

Subtract 2y from both sides:

${ 5x = -7 - 2y }$

Divide both sides by 5:

${ x = \frac{-7 - 2y}{5} }$

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

${ 15\left(\frac{-7 - 2y}{5}\right) + 6y = 21 }$

Expand and simplify the equation:

${ -21 - 6y + 6y = 21 }$

${ -21 = 21 }$

This equation is false for all values of y. Therefore, the system has no solution.

Conclusion

In conclusion, both systems have no solution. The first system has a contradiction, and the second system has a false equation. Therefore, the correct answer is:

A. Both systems have no solution.

Why is it Important to Determine the Number of Solutions?

Determining the number of solutions is important because it helps us understand the relationship between the equations in the system. If a system has no solution, it means that the equations are inconsistent, and there is no value of the variables that satisfies all the equations. On the other hand, if a system has an infinite number of solutions, it means that the equations are dependent, and there are many values of the variables that satisfy all the equations.

How to Determine the Number of Solutions?

To determine the number of solutions, we can use the following steps:

Solve one of the equations for one of the variables. This will give us an expression for one of the variables in terms of the other variable.
Substitute the expression into the other equation. This will give us an equation in one variable.
Determine if the equation is true or false. If the equation is true, it means that the system has an infinite number of solutions. If the equation is false, it means that the system has no solution.

Real-World Applications

Determining the number of solutions has many real-world applications. For example, in economics, a system of linear equations can be used to model the relationship between different economic variables, such as supply and demand. In engineering, a system of linear equations can be used to model the behavior of a physical system, such as a circuit or a mechanical system.

Final Thoughts

In conclusion, determining the number of solutions is an important step in solving a system of linear equations. By following the steps outlined above, we can determine whether a system has an infinite number of solutions, no solution, or a unique solution. This knowledge can be used to make informed decisions in many real-world applications.

References

[1] Linear Algebra and Its Applications by Gilbert Strang
[2] Introduction to Linear Algebra by Jim Hefferon
[3] Solving Systems of Linear Equations by Math Open Reference

Additional Resources

[1] Linear Algebra Tutorial by Khan Academy
[2] Solving Systems of Linear Equations by MIT OpenCourseWare
[3] Linear Algebra and Its Applications by Wolfram MathWorld

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations means finding the values of the variables that satisfy all the equations in the system.

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

A: To determine the number of solutions, you can use the following steps:

Solve one of the equations for one of the variables. This will give you an expression for one of the variables in terms of the other variable.
Substitute the expression into the other equation. This will give you an equation in one variable.
Determine if the equation is true or false. If the equation is true, it means that the system has an infinite number of solutions. If the equation is false, it means that the system has no solution.

Q: What is the difference between an identity and a contradiction?

A: An identity is an equation that is true for all values of the variables. A contradiction is an equation that is false for all values of the variables.

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

A: If the system has an identity, it means that the system has an infinite number of solutions. If the system has a contradiction, it means that the system has no solution.

Q: Can a system have a unique solution?

A: Yes, a system can have a unique solution. This occurs when the system has a single solution that satisfies all the equations.

Q: How do I find the unique solution?

A: To find the unique solution, you can use the following steps:

Solve one of the equations for one of the variables. This will give you an expression for one of the variables in terms of the other variable.
Substitute the expression into the other equation. This will give you an equation in one variable.
Solve the equation for the variable. This will give you the value of the variable.
Substitute the value into one of the original equations. This will give you the value of the other variable.

Q: What are some real-world applications of determining the number of solutions?

A: Determining the number of solutions has many real-world applications, including:

Economics: A system of linear equations can be used to model the relationship between different economic variables, such as supply and demand.
Engineering: A system of linear equations can be used to model the behavior of a physical system, such as a circuit or a mechanical system.
Computer Science: A system of linear equations can be used to solve problems in computer science, such as finding the shortest path in a graph.

Q: What are some common mistakes to avoid when determining the number of solutions?

A: Some common mistakes to avoid when determining the number of solutions include:

Not following the steps correctly. Make sure to solve one of the equations for one of the variables, substitute the expression into the other equation, and determine if the equation is true or false.
Not checking for identities and contradictions. Make sure to check if the system has an identity or a contradiction, as this can affect the number of solutions.
Not using the correct method. Make sure to use the correct method to determine the number of solutions, such as using the steps outlined above.

Q: What are some resources for learning more about determining the number of solutions?

A: Some resources for learning more about determining the number of solutions include:

Linear Algebra Tutorial by Khan Academy
Solving Systems of Linear Equations by MIT OpenCourseWare
Linear Algebra and Its Applications by Wolfram MathWorld

Q: Can I use a calculator to determine the number of solutions?

A: Yes, you can use a calculator to determine the number of solutions. Many calculators have built-in functions for solving systems of linear equations, such as the "solve" function.

Q: How do I know if a system has a unique solution or an infinite number of solutions?

A: If the system has a unique solution, it means that there is only one value of the variables that satisfies all the equations. If the system has an infinite number of solutions, it means that there are many values of the variables that satisfy all the equations.

Q: Can a system have both a unique solution and an infinite number of solutions?

A: No, a system cannot have both a unique solution and an infinite number of solutions. If a system has a unique solution, it means that there is only one value of the variables that satisfies all the equations. If a system has an infinite number of solutions, it means that there are many values of the variables that satisfy all the equations.

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

A: To determine the number of solutions for a system of linear equations with three variables, you can use the following steps:

Solve one of the equations for one of the variables. This will give you an expression for one of the variables in terms of the other two variables.
Substitute the expression into the other two equations. This will give you two equations in two variables.
Determine if the equations are true or false. If the equations are true, it means that the system has an infinite number of solutions. If the equations are false, it means that the system has no solution.

Q: Can a system of linear equations have more than three variables?

A: Yes, a system of linear equations can have more than three variables. However, determining the number of solutions for a system with more than three variables can be more complex and may require the use of advanced mathematical techniques.

Q: How do I determine the number of solutions for a system of linear equations with more than three variables?

A: To determine the number of solutions for a system of linear equations with more than three variables, you can use the following steps:

Solve one of the equations for one of the variables. This will give you an expression for one of the variables in terms of the other variables.
Substitute the expression into the other equations. This will give you a system of equations in fewer variables.
Determine if the system has an identity or a contradiction. If the system has an identity, it means that the system has an infinite number of solutions. If the system has a contradiction, it means that the system has no solution.

Q: What are some common mistakes to avoid when determining the number of solutions for a system of linear equations with more than three variables?

A: Some common mistakes to avoid when determining the number of solutions for a system of linear equations with more than three variables include:

Not following the steps correctly. Make sure to solve one of the equations for one of the variables, substitute the expression into the other equations, and determine if the system has an identity or a contradiction.
Not checking for identities and contradictions. Make sure to check if the system has an identity or a contradiction, as this can affect the number of solutions.
Not using the correct method. Make sure to use the correct method to determine the number of solutions, such as using the steps outlined above.

Q: What are some resources for learning more about determining the number of solutions for a system of linear equations with more than three variables?

A: Some resources for learning more about determining the number of solutions for a system of linear equations with more than three variables include:

Linear Algebra Tutorial by Khan Academy
Solving Systems of Linear Equations by MIT OpenCourseWare
Linear Algebra and Its Applications by Wolfram MathWorld