Select The Correct Answer.A System Of Equations And Its Solution Are Given Below.System A $ \begin{array}{c} 5x - Y = -11 \ 3x - 2y = -8 \ \text{Solution (-2, 1) \end{array} }$To Get System { B $}$ Below, The Second

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. These systems can be linear or non-linear, and they can have one or more solutions. In this article, we will discuss a system of equations and its solution, and we will explore how to get a new system of equations from the original one.

System A

The system of equations A is given by:

$$\begin{array}{c} 5x - y = -11 \\ 3x - 2y = -8 \\ \text{Solution: } (-2, 1) \end{array}$$

This system consists of two linear equations with two variables, x and y. The solution to this system is given as (-2, 1), which means that when x = -2 and y = 1, both equations are satisfied.

Understanding the Solution

To understand the solution, let's substitute the values of x and y into both equations.

For the first equation, we have:

$$5(-2) - 1 = -11$$

Expanding and simplifying, we get:

$$-10 - 1 = -11$$

$$-11 = -11$$

This shows that the first equation is satisfied when x = -2 and y = 1.

For the second equation, we have:

$$3(-2) - 2(1) = -8$$

Expanding and simplifying, we get:

$$-6 - 2 = -8$$

$$-8 = -8$$

This shows that the second equation is also satisfied when x = -2 and y = 1.

Getting System B

The problem states that to get system B, we need to modify the second equation of system A. Let's analyze the second equation:

$$3x - 2y = -8$$

We can multiply this equation by a constant to get a new equation. Let's multiply it by 2:

$$6x - 4y = -16$$

This is a new equation that is similar to the second equation of system A, but with a different constant.

System B

The new system of equations B is given by:

$$\begin{array}{c} 5x - y = -11 \\ 6x - 4y = -16 \\ \text{Solution: } (-2, 1) \end{array}$$

This system consists of two linear equations with two variables, x and y. The solution to this system is still given as (-2, 1), which means that when x = -2 and y = 1, both equations are satisfied.

Comparing System A and System B

Let's compare the two systems:

System A:

$$\begin{array}{c} 5x - y = -11 \\ 3x - 2y = -8 \\ \text{Solution: } (-2, 1) \end{array}$$

System B:

$$\begin{array}{c} 5x - y = -11 \\ 6x - 4y = -16 \\ \text{Solution: } (-2, 1) \end{array}$$

The only difference between the two systems is the second equation. In system A, the second equation is:

$$3x - 2y = -8$$

In system B, the second equation is:

$$6x - 4y = -16$$

This shows that we can modify the second equation of system A to get system B.

Conclusion

In this article, we discussed a system of equations and its solution, and we explored how to get a new system of equations from the original one. We analyzed the solution to the original system and showed that it is still satisfied when we modify the second equation. This demonstrates that we can create new systems of equations by modifying the original equations.

Key Takeaways

• A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.
• The solution to a system of equations is a set of values that satisfy all the equations in the system.
• We can modify the equations in a system of equations to create a new system of equations.
• The solution to the new system of equations may be the same as the solution to the original system, or it may be different.

Further Reading

If you want to learn more about systems of equations and their solutions, I recommend checking out the following resources:

• Khan Academy: Systems of Equations
• Mathway: Systems of Equations
• Wolfram Alpha: Systems of Equations

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the 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 equations are consistent (i.e., they have the same solution).
• Use the method of substitution or elimination to solve the system.
• Use a graphing calculator or computer software to visualize the solution.

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

A: A system of linear equations consists of two or more linear equations, while a system of non-linear equations consists of two or more non-linear equations. Linear equations are equations in which the variables are raised to the power of 1, while non-linear equations are equations in which the variables are raised to a power other than 1.

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

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

• Method of substitution: Substitute the expression for one variable from one equation into the other equation.
• Method of elimination: Add or subtract the equations to eliminate one variable.
• Graphing method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the solution to a system of equations?

A: The solution to a system of equations is a set of values that satisfy all the equations in the system. This can be a single point, a line, or a plane, depending on the number of variables and equations.

Q: How do I check if a solution is correct?

A: To check if a solution is correct, you can substitute the values into each equation and verify that they are true.

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

A: A dependent system of equations is a system in which the equations are not independent, meaning that one equation can be expressed as a multiple of the other. An independent system of equations is a system in which the equations are not dependent, meaning that they have a unique solution.

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

A: To determine if a system of equations is dependent or independent, you can use the following methods:

• Check if the equations are multiples of each other.
• Use the method of substitution or elimination to solve the system.
• Use a graphing calculator or computer software to visualize the solution.

Q: What is the significance of a system of equations in real-world applications?

A: Systems of equations are used in a wide range of real-world applications, including:

• Physics: To describe the motion of objects and the forces acting on them.
• Engineering: To design and optimize systems, such as bridges and buildings.
• Economics: To model the behavior of markets and economies.
• Computer Science: To solve problems in computer graphics, game development, and artificial intelligence.

Q: How can I practice solving systems of equations?

A: You can practice solving systems of equations by:

• Working through examples and exercises in a textbook or online resource.
• Using a graphing calculator or computer software to visualize the solution.
• Creating your own systems of equations and solving them.
• Joining a study group or online community to work through problems with others.

Conclusion

In this article, we have answered some of the most frequently asked questions about systems of equations. We have covered topics such as the definition of a system of equations, how to determine if a system has a solution, and how to solve a system of linear equations. We have also discussed the significance of systems of equations in real-world applications and provided tips for practicing solving systems of equations.