Substituting the Constraints

Oct 13, 2025 by talynt.org 46 views

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Introduction

In linear algebra and systems of equations, finding the solution to a system of equations is a fundamental problem. However, in some cases, we are given additional constraints that need to be satisfied. In this article, we will explore how to find the solution of a system of equations when we are given that $x_1=0$ and $s_1=0$ . We will use the given system of equations and the constraints to find the solution.

The System of Equations

The given system of equations is:

\begin{aligned} 3x_1 + 4x_2 + s_1 & = 8 \\ x_1 + 5x_2 + s_2 & = 17 \end{aligned}

We are given that $x_1=0$ and $s_1=0$ . We need to find the solution of the system of equations satisfying these constraints.

Substituting the Constraints

We will substitute the given constraints into the system of equations. Since $x_1=0$ , we can substitute $x_1=0$ into the first equation:

3(0) + 4x_2 + 0 = 8

Simplifying the equation, we get:

4x_2 = 8

Dividing both sides by 4, we get:

x_2 = 2

Finding the Value of $s_2$

We will substitute the value of $x_2$ into the second equation:

0 + 5(2) + s_2 = 17

Simplifying the equation, we get:

10 + s_2 = 17

Subtracting 10 from both sides, we get:

s_2 = 7

The Basic Solution

The basic solution of the system of equations is given by:

\left(x_1, x_2, s_1, s_2\right) = (0, 2, 0, 7)

Conclusion

In this article, we found the solution of a system of equations with given constraints. We substituted the constraints into the system of equations and solved for the variables. The basic solution of the system of equations is given by $(0, 2, 0, 7)$ .

Example Use Cases

This problem can be used in various fields such as:

Linear Algebra: This problem is a fundamental example of solving a system of equations with constraints.
Computer Science: This problem can be used to illustrate the concept of constraints in programming.
Mathematics: This problem can be used to teach students about solving systems of equations with constraints.

Step-by-Step Solution

Here are the steps to solve the problem:

Substitute the constraints into the system of equations: Substitute $x_1=0$ and $s_1=0$ into the system of equations.
Solve for $x_2$ : Solve the resulting equation for $x_2$ .
Find the value of $s_2$ : Substitute the value of $x_2$ into the second equation and solve for $s_2$ .
Write the basic solution: Write the basic solution of the system of equations.

Tips and Tricks

Here are some tips and tricks to solve the problem:

Use substitution: Substitute the constraints into the system of equations to simplify the problem.
Solve for one variable at a time: Solve for one variable at a time to avoid confusion.
Check your work: Check your work to ensure that the solution satisfies the constraints.

Common Mistakes

Here are some common mistakes to avoid:

Not substituting the constraints: Not substituting the constraints into the system of equations can lead to incorrect solutions.
Not solving for one variable at a time: Not solving for one variable at a time can lead to confusion and incorrect solutions.
Not checking your work: Not checking your work can lead to incorrect solutions.

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Q: What is the basic solution of the system of equations?

A: The basic solution of the system of equations is given by $(0, 2, 0, 7)$ .

Q: How do I find the solution of a system of equations with given constraints?

A: To find the solution of a system of equations with given constraints, you need to substitute the constraints into the system of equations and solve for the variables.

Q: What are the steps to solve the problem?

A: The steps to solve the problem are:

Substitute the constraints into the system of equations: Substitute $x_1=0$ and $s_1=0$ into the system of equations.
Solve for $x_2$ : Solve the resulting equation for $x_2$ .
Find the value of $s_2$ : Substitute the value of $x_2$ into the second equation and solve for $s_2$ .
Write the basic solution: Write the basic solution of the system of equations.

Q: What are some common mistakes to avoid?

A: Some common mistakes to avoid are:

Not substituting the constraints: Not substituting the constraints into the system of equations can lead to incorrect solutions.
Not solving for one variable at a time: Not solving for one variable at a time can lead to confusion and incorrect solutions.
Not checking your work: Not checking your work can lead to incorrect solutions.

Q: What are some tips and tricks to solve the problem?

A: Some tips and tricks to solve the problem are:

Use substitution: Substitute the constraints into the system of equations to simplify the problem.
Solve for one variable at a time: Solve for one variable at a time to avoid confusion.
Check your work: Check your work to ensure that the solution satisfies the constraints.

Q: What are some example use cases of this problem?

A: This problem can be used in various fields such as:

Linear Algebra: This problem is a fundamental example of solving a system of equations with constraints.
Computer Science: This problem can be used to illustrate the concept of constraints in programming.
Mathematics: This problem can be used to teach students about solving systems of equations with constraints.

Q: How do I check my work?

A: To check your work, you need to ensure that the solution satisfies the constraints. In this case, we need to check that $x_1=0$ and $s_1=0$ .

Q: What if I get stuck?

A: If you get stuck, you can try the following:

Re-read the problem: Re-read the problem to ensure that you understand what is being asked.
Check your work: Check your work to ensure that you have not made any mistakes.
Ask for help: Ask for help from a teacher, tutor, or classmate.

Q: Can I use this problem in my own work?

A: Yes, you can use this problem in your own work. However, be sure to give credit to the original source of the problem.

Q: Is this problem suitable for all levels of students?

A: This problem is suitable for students who have a basic understanding of linear algebra and systems of equations. However, it may be challenging for students who are new to these concepts.