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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. However, in some cases, a system of linear equations may not have a unique solution, but rather an infinite number of solutions. In this article, we will explore how to solve a system of linear equations with infinite solutions.

The System of Linear Equations

The given system of linear equations is:

x−8y+z=6y−8z=−5\begin{align*} x - 8y + z &= 6 \\ y - 8z &= -5 \end{align*}

To solve this system, we can use the method of substitution or elimination. However, since the system has infinite solutions, we will use a different approach.

Solving the System

Let's start by solving the second equation for y:

y=8z+5y = 8z + 5

Now, substitute this expression for y into the first equation:

x−8(8z+5)+z=6x - 8(8z + 5) + z = 6

Expand and simplify the equation:

x−64z−40+z=6x - 64z - 40 + z = 6

Combine like terms:

x−63z−40=6x - 63z - 40 = 6

Add 40 to both sides:

x−63z=46x - 63z = 46

Now, we have a new equation with two variables, x and z. However, we can see that this equation is not enough to determine the values of x and z uniquely. In fact, this equation is true for any value of x and z that satisfies the original system.

The Infinite Solutions

To find the infinite solutions, we can use the fact that the second equation is a linear combination of the first equation. Specifically, we can multiply the first equation by 8 and add it to the second equation:

8(x−8y+z)+(y−8z)=8(6)+(−5)8(x - 8y + z) + (y - 8z) = 8(6) + (-5)

Expand and simplify the equation:

8x−64y+8z+y−8z=48−58x - 64y + 8z + y - 8z = 48 - 5

Combine like terms:

8x−63y=438x - 63y = 43

Now, we have a new equation with two variables, x and y. However, we can see that this equation is not enough to determine the values of x and y uniquely. In fact, this equation is true for any value of x and y that satisfies the original system.

Conclusion

In conclusion, the given system of linear equations has infinite solutions. We can see that the second equation is a linear combination of the first equation, and by multiplying the first equation by 8 and adding it to the second equation, we get a new equation with two variables, x and y. However, this equation is not enough to determine the values of x and y uniquely, and we can see that it is true for any value of x and y that satisfies the original system.

The Final Answer

The final answer is that the system of linear equations has infinite solutions. We can see that the values of x and z can be any real numbers that satisfy the equation x - 63z = 46, and the values of y can be any real number that satisfies the equation y = 8z + 5.

Discussion

The given system of linear equations is a classic example of a system with infinite solutions. In this system, we can see that the second equation is a linear combination of the first equation, and by multiplying the first equation by 8 and adding it to the second equation, we get a new equation with two variables, x and y. However, this equation is not enough to determine the values of x and y uniquely, and we can see that it is true for any value of x and y that satisfies the original system.

References

  • [1] "Linear Algebra and Its Applications" by Gilbert Strang
  • [2] "Introduction to Linear Algebra" by Jim Hefferon

Note

Q: What is a system of linear equations with infinite solutions?

A: A system of linear equations with infinite solutions is a set of two or more equations that involve two or more variables, where the system has an infinite number of solutions. This means that there are an infinite number of values of the variables that satisfy all the equations in the system.

Q: How can I determine if a system of linear equations has infinite solutions?

A: To determine if a system of linear equations has infinite solutions, you can use the following methods:

  • Check if the system is inconsistent, meaning that there is no solution that satisfies all the equations.
  • Check if the system is dependent, meaning that one equation is a multiple of another equation.
  • Check if the system has a free variable, meaning that one variable can take on any value and still satisfy all the equations.

Q: What is a free variable?

A: A free variable is a variable in a system of linear equations that can take on any value and still satisfy all the equations. In other words, a free variable is a variable that is not constrained by the equations in the system.

Q: How can I solve a system of linear equations with infinite solutions?

A: To solve a system of linear equations with infinite solutions, you can use the following methods:

  • Use the method of substitution or elimination to solve the system.
  • Use the method of matrices to solve the system.
  • Use the method of parametric equations to solve the system.

Q: What is the method of parametric equations?

A: The method of parametric equations is a method of solving a system of linear equations by expressing the variables in terms of a parameter. This method is useful for solving systems with infinite solutions.

Q: How do I use the method of parametric equations to solve a system of linear equations with infinite solutions?

A: To use the method of parametric equations to solve a system of linear equations with infinite solutions, follow these steps:

  1. Express one variable in terms of a parameter.
  2. Substitute the expression for the variable into the other equations in the system.
  3. Solve the resulting system of equations for the parameter.
  4. Express the variables in terms of the parameter.

Q: What are some examples of systems of linear equations with infinite solutions?

A: Some examples of systems of linear equations with infinite solutions include:

  • The system of equations x - 8y + z = 6 and y - 8z = -5.
  • The system of equations 2x + 3y = 5 and x + 2y = 3.
  • The system of equations x + 2y = 4 and 2x + 4y = 8.

Q: How can I use technology to solve a system of linear equations with infinite solutions?

A: You can use technology such as graphing calculators or computer algebra systems to solve a system of linear equations with infinite solutions. These tools can help you visualize the system and find the solutions.

Q: What are some common mistakes to avoid when solving a system of linear equations with infinite solutions?

A: Some common mistakes to avoid when solving a system of linear equations with infinite solutions include:

  • Assuming that the system has a unique solution.
  • Failing to check for free variables.
  • Failing to use the correct method of solution.

Q: How can I practice solving systems of linear equations with infinite solutions?

A: You can practice solving systems of linear equations with infinite solutions by working through examples and exercises in a textbook or online resource. You can also try solving systems with infinite solutions on your own using the methods described in this article.