Solve systems of linear equations

Oct 13, 2025 by miodeya.com 70 views

Introduction

In mathematics, solving systems of linear equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. This article will focus on determining which pair of equations has the point ${(-3, 2)}$ as a solution.

Understanding the Problem

To solve this problem, we need to understand the concept of a solution to a system of linear equations. A solution to a system of linear equations is an ordered pair of values that satisfies all the equations in the system. In this case, we are given a point ${(-3, 2)}$ and we need to determine which pair of equations has this point as a solution.

Pair A: $${2x + 4y = -2}$ and $${-2x + 2y = 10}$

Let's start by analyzing Pair A. We can substitute the values of x and y from the given point ${(-3, 2)}$ into the first equation:

{2(-3) + 4(2) = -2\}

Simplifying the equation, we get:

{-6 + 8 = -2\}

{2 = -2\}

This equation is not true, so the point ${(-3, 2)}$ is not a solution to Pair A.

Pair B: $${2x + 4y = 2}$ and $${-2x + 2y = -10}$

Next, let's analyze Pair B. We can substitute the values of x and y from the given point ${(-3, 2)}$ into the first equation:

{2(-3) + 4(2) = 2\}

Simplifying the equation, we get:

{-6 + 8 = 2\}

{2 = 2\}

This equation is true, so the point ${(-3, 2)}$ is a solution to the first equation in Pair B.

Now, let's substitute the values of x and y from the given point ${(-3, 2)}$ into the second equation:

{-2(-3) + 2(2) = -10\}

Simplifying the equation, we get:

{6 + 4 = -10\}

{10 = -10\}

This equation is not true, so the point ${(-3, 2)}$ is not a solution to Pair B.

Pair C: $${2x + 4y = -2}$ and $${-2x + 2y = 10}$

Finally, let's analyze Pair C. We can substitute the values of x and y from the given point ${(-3, 2)}$ into the first equation:

{2(-3) + 4(2) = -2\}

Simplifying the equation, we get:

{-6 + 8 = -2\}

{2 = -2\}

This equation is not true, so the point ${(-3, 2)}$ is not a solution to Pair C.

Conclusion

Why is this important?

Understanding which pair of equations has a given point as a solution is important in mathematics because it helps us to:

Solve systems of linear equations: By identifying which pair of equations has a given point as a solution, we can determine the values of variables that satisfy multiple equations simultaneously.
Analyze the behavior of systems: By analyzing which pair of equations has a given point as a solution, we can gain insights into the behavior of the system and make predictions about the values of variables.
Make informed decisions: By understanding which pair of equations has a given point as a solution, we can make informed decisions about the values of variables and the behavior of the system.

Real-world applications

The concept of solving systems of linear equations has numerous real-world applications, including:

Physics and engineering: Solving systems of linear equations is essential in physics and engineering to model and analyze complex systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems, such as supply and demand curves.
Computer science: Solving systems of linear equations is used in computer science to solve problems in computer graphics, machine learning, and data analysis.

Final thoughts

$In conclusion, the point ${(-3, 2)}$ is a solution to Pair B: $${2x + 4y = 2\}$ and $${-2x + 2y = -10\}$$. This is because the point satisfies the first equation in Pair B, and the second equation in Pair B is not true, which means that the point is not a solution to Pair B. Understanding which pair of equations has a given point as a solution is important in mathematics because it helps us to solve systems of linear equations, analyze the behavior of systems, and make informed decisions. The concept of solving systems of linear equations has numerous real-world applications, including physics and engineering, economics, and computer science.<br/> **The Point ${(-3, 2)}$ is a Solution to Which of the Following Pairs of Equations? - Q&A**$

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

A: A solution to a system of linear equations is an ordered pair of values that satisfies all the equations in the system.

Q: How do you determine which pair of equations has a given point as a solution?

A: To determine which pair of equations has a given point as a solution, you need to substitute the values of x and y from the given point into each equation and check if the equation is true.

Q: What is the significance of solving systems of linear equations?

A: Solving systems of linear equations is significant because it helps us to:

Solve systems of linear equations: By identifying which pair of equations has a given point as a solution, we can determine the values of variables that satisfy multiple equations simultaneously.
Analyze the behavior of systems: By analyzing which pair of equations has a given point as a solution, we can gain insights into the behavior of the system and make predictions about the values of variables.
Make informed decisions: By understanding which pair of equations has a given point as a solution, we can make informed decisions about the values of variables and the behavior of the system.

Q: What are some real-world applications of solving systems of linear equations?

A: The concept of solving systems of linear equations has numerous real-world applications, including:

Physics and engineering: Solving systems of linear equations is essential in physics and engineering to model and analyze complex systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used in economics to model and analyze economic systems, such as supply and demand curves.
Computer science: Solving systems of linear equations is used in computer science to solve problems in computer graphics, machine learning, and data analysis.

Q: How do you know which pair of equations has a given point as a solution?

A: To know which pair of equations has a given point as a solution, you need to substitute the values of x and y from the given point into each equation and check if the equation is true. If the equation is true, then the point is a solution to that equation.

Q: What is the importance of understanding which pair of equations has a given point as a solution?

A: Understanding which pair of equations has a given point as a solution is important because it helps us to:

Solve systems of linear equations: By identifying which pair of equations has a given point as a solution, we can determine the values of variables that satisfy multiple equations simultaneously.
Analyze the behavior of systems: By analyzing which pair of equations has a given point as a solution, we can gain insights into the behavior of the system and make predictions about the values of variables.
Make informed decisions: By understanding which pair of equations has a given point as a solution, we can make informed decisions about the values of variables and the behavior of the system.

Q: Can you provide an example of a system of linear equations with a given point as a solution?

A: Yes, consider the following system of linear equations:

{2x + 4y = 2\}

{-2x + 2y = -10\}

The point ${(-3, 2)}$ is a solution to this system of linear equations because it satisfies both equations.

Q: How do you determine the values of variables that satisfy multiple equations simultaneously?

A: To determine the values of variables that satisfy multiple equations simultaneously, you need to solve the system of linear equations. This can be done using various methods, such as substitution, elimination, or graphing.

Q: What are some common methods for solving systems of linear equations?

A: Some common methods for solving systems of linear equations include:

Substitution method: This involves substituting the values of one variable from one equation into the other equation to solve for the other variable.
Elimination method: This involves adding or subtracting the equations to eliminate one variable and solve for the other variable.
Graphing method: This involves graphing the equations on a coordinate plane and finding the point of intersection, which represents the solution to the system.

Q: Can you provide an example of using the substitution method to solve a system of linear equations?

A: Yes, consider the following system of linear equations:

{2x + 4y = 2\}

{-2x + 2y = -10\}

We can use the substitution method to solve this system by substituting the values of x from the first equation into the second equation:

{-2(-3) + 2y = -10\}

Simplifying the equation, we get:

{6 + 2y = -10\}

{2y = -16\}

{y = -8\}

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

{2x + 4(-8) = 2\}

Simplifying the equation, we get:

{2x - 32 = 2\}

{2x = 34\}

{x = 17\}

Therefore, the solution to the system of linear equations is x = 17 and y = -8.

Q: Can you provide an example of using the elimination method to solve a system of linear equations?

A: Yes, consider the following system of linear equations:

{2x + 4y = 2\}

{-2x + 2y = -10\}

We can use the elimination method to solve this system by adding the two equations to eliminate the x variable:

{2x + 4y + (-2x + 2y) = 2 + (-10)\}

Simplifying the equation, we get:

{6y = -8\}

{y = -\frac{4}{3}\}

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

{2x + 4(-\frac{4}{3}) = 2\}

Simplifying the equation, we get:

{2x - \frac{16}{3} = 2\}

{2x = \frac{10}{3}\}

{x = \frac{5}{3}\}

Therefore, the solution to the system of linear equations is x = 5/3 and y = -4/3.

Q: Can you provide an example of using the graphing method to solve a system of linear equations?

A: Yes, consider the following system of linear equations:

{2x + 4y = 2\}

{-2x + 2y = -10\}

We can use the graphing method to solve this system by graphing the equations on a coordinate plane and finding the point of intersection, which represents the solution to the system.

To graph the first equation, we can use the slope-intercept form:

{y = -\frac{1}{2}x + \frac{1}{2}\}

To graph the second equation, we can use the slope-intercept form:

{y = x - 5\}

By graphing the two equations on a coordinate plane, we can see that they intersect at the point (17, -8).

Therefore, the solution to the system of linear equations is x = 17 and y = -8.

Conclusion

In conclusion, the point ${(-3, 2)}$ is a solution to Pair B: $${2x + 4y = 2}$ and $${-2x + 2y = -10}$$. This is because the point satisfies the first equation in Pair B, and the second equation in Pair B is not true, which means that the point is not a solution to Pair B. Understanding which pair of equations has a given point as a solution is important in mathematics because it helps us to solve systems of linear equations, analyze the behavior of systems, and make informed decisions. The concept of solving systems of linear equations has numerous real-world applications, including physics and engineering, economics, and computer science.