Two Systems Of Equations Are Shown. The First Equation In System B Is The Original Equation In System A. The Second Equation In System B Is The Sum Of That Equation And A Multiple Of The Second Equation In System A.System A:1. [$\frac{1}{2} X + 3y

Introduction

Systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore two systems of equations, and we will discuss how to solve them using a step-by-step approach.

System A

The first system of equations is:

1. $\frac{1}{2} x + 3y = 7$
2. $x - 2y = -3$

System B

The second system of equations is:

1. $\frac{1}{2} x + 3y = 7$
2. $(\frac{1}{2} x + 3y) + 2(x - 2y) = 7 + 2(-3)$

Simplifying System B

Let's simplify the second equation in System B by combining like terms:

$(\frac{1}{2} x + 3y) + 2(x - 2y) = 7 + 2(-3)$

$\frac{1}{2} x + 3y + 2x - 4y = 7 - 6$

$\frac{5}{2} x - y = 1$

Comparing Systems A and B

Now, let's compare the two systems of equations. We can see that the first equation in System B is the same as the first equation in System A. The second equation in System B is the sum of the first equation in System A and a multiple of the second equation in System A.

The Multiple

The multiple is 2, which is the coefficient of the second equation in System A. This means that we are multiplying the second equation in System A by 2 and adding it to the first equation in System A.

Why This Works

This method works because the second equation in System B is a linear combination of the two equations in System A. By adding a multiple of one equation to another, we are creating a new equation that is a linear combination of the original equations.

Solving System B

Now that we have simplified System B, let's solve it using the same method as before. We can use the substitution method or the elimination method to solve the system.

Substitution Method

Let's use the substitution method to solve the system. We can solve the first equation for x:

$\frac{1}{2} x + 3y = 7$

$\frac{1}{2} x = 7 - 3y$

$x = 2(7 - 3y)$

$x = 14 - 6y$

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

$(\frac{1}{2} x + 3y) + 2(x - 2y) = 7 + 2(-3)$

$\frac{1}{2} (14 - 6y) + 3y + 2(14 - 6y) - 4y = 7 - 6$

$7 - 3y + 6y + 28 - 12y - 4y = 1$

$35 - 19y = 1$

$-19y = -34$

$y = \frac{34}{19}$

Now, we can substitute this value of y back into one of the original equations to solve for x:

$\frac{1}{2} x + 3y = 7$

$\frac{1}{2} x + 3(\frac{34}{19}) = 7$

$\frac{1}{2} x + \frac{102}{19} = 7$

$\frac{1}{2} x = 7 - \frac{102}{19}$

$\frac{1}{2} x = \frac{133 - 102}{19}$

$\frac{1}{2} x = \frac{31}{19}$

$x = \frac{62}{19}$

Conclusion

In this article, we have explored two systems of equations and discussed how to solve them using a step-by-step approach. We have seen how to simplify a system of equations by combining like terms and how to use the substitution method to solve the system. We have also seen how to compare two systems of equations and how to use the elimination method to solve the system.

The Importance of Systems of Equations

Systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. Systems of equations are used in a wide range of fields, including physics, engineering, economics, and computer science. By understanding how to solve systems of equations, we can gain a deeper understanding of the world around us and make more informed decisions.

Real-World Applications

Systems of equations have many real-world applications. For example, in physics, systems of equations are used to describe the motion of objects and to predict the behavior of complex systems. In engineering, systems of equations are used to design and optimize systems, such as bridges and buildings. In economics, systems of equations are used to model the behavior of markets and to predict the impact of policy changes.

Conclusion

In conclusion, systems of equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding how to solve systems of equations, we can gain a deeper understanding of the world around us and make more informed decisions. We hope that this article has provided a helpful introduction to systems of equations and has inspired readers to learn more about this important topic.

Additional Resources

For additional resources on systems of equations, we recommend the following:

• Khan Academy: Systems of Equations
• MIT OpenCourseWare: Linear Algebra
• Wolfram MathWorld: Systems of Equations

Final Thoughts

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other through a common variable or variables.

Q: How do I know if a system of equations has a solution?

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

• Check if the two equations are parallel (i.e., they have the same slope but different y-intercepts). If they are parallel, the system has no solution.
• Check if the two equations are identical. If they are identical, the system has infinitely many solutions.
• Check if the two equations intersect at a single point. If they intersect at a single point, the system has a unique solution.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including:

• Substitution method: Substitute one equation into the other equation to solve for the variable.
• Elimination method: Add or subtract the two equations to eliminate one variable.
• Graphical method: Graph the two equations on a coordinate plane and find the point of intersection.

Q: What is the difference between a system of linear equations and a system of nonlinear equations?

A: A system of linear equations is a set of equations where each equation is a linear equation (i.e., it can be written in the form ax + by = c). A system of nonlinear equations is a set of equations where at least one equation is a nonlinear equation (i.e., it cannot be written in the form ax + by = c).

Q: Can a system of equations have more than one solution?

A: Yes, a system of equations can have more than one solution. This occurs when the two equations are identical or when the system has infinitely many solutions.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution. This occurs when the two equations are parallel or when the system has no intersection point.

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

A: A system of equations is consistent if it has at least one solution. A system of equations is inconsistent if it has no solution.

Q: What is the difference between a consistent system of equations and an inconsistent system of equations?

A: A consistent system of equations is a system that has at least one solution. An inconsistent system of equations is a system that has no solution.

Q: Can a system of equations be both consistent and inconsistent?

A: No, a system of equations cannot be both consistent and inconsistent. A system of equations is either consistent or inconsistent.

Q: How do I determine if a system of equations is dependent or independent?

A: A system of equations is dependent if the two equations are identical or if one equation is a multiple of the other equation. A system of equations is independent if the two equations are not identical and one equation is not a multiple of the other equation.

Q: What is the difference between a dependent system of equations and an independent system of equations?

A: A dependent system of equations is a system where the two equations are identical or one equation is a multiple of the other equation. An independent system of equations is a system where the two equations are not identical and one equation is not a multiple of the other equation.

Q: Can a system of equations be both dependent and independent?

A: No, a system of equations cannot be both dependent and independent. A system of equations is either dependent or independent.

Conclusion

In this article, we have answered some of the most frequently asked questions about systems of equations. We hope that this article has provided a helpful resource for students and professionals who are learning about systems of equations. If you have any further questions, please don't hesitate to ask.