What Is The Solution To This System Of Linear Equations?${ \begin{align} 7x - 2y &= -6 \ 8x + Y &= 3 \end{align} }$A. { (-6, 3)$}$B. { (0, 3)$}$C. { (1, -5)$}$D. { (15, -1)$}$

Mar 7, 2025 by ADMIN 179 views

**Solving a System of Linear Equations: A Step-by-Step Guide**

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. Solving a system of linear equations is a fundamental concept in algebra and is used extensively in various fields such as physics, engineering, economics, and computer science. In this article, we will focus on solving a system of two linear equations with two variables.

What is a System of Linear Equations?

A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables. Each equation in the system is a linear equation, which means that it can be written in the form:

ax + by = c

where a, b, and c are constants, and x and y are variables.

Example of a System of Linear Equations

Let's consider the following system of linear equations:

7x - 2y = -6 8x + y = 3

This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.

Methods for Solving a System of Linear Equations

There are several methods for solving a system of linear equations, including:

Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Substitution Method

Let's use the substitution method to solve the system of linear equations.

Step 1: Solve the First Equation for x

We can solve the first equation for x by adding 2y to both sides:

7x = -6 + 2y

Then, we can divide both sides by 7:

x = (-6 + 2y) / 7

Step 2: Substitute the Expression for x into the Second Equation

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

8((-6 + 2y) / 7) + y = 3

Step 3: Simplify the Equation

We can simplify the equation by multiplying both sides by 7:

8(-6 + 2y) + 7y = 21

Expanding the left-hand side, we get:

-48 + 16y + 7y = 21

Combine like terms:

-48 + 23y = 21

Add 48 to both sides:

23y = 69

Divide both sides by 23:

y = 3

Step 4: Find the Value of x

Now that we have the value of y, we can find the value of x by substituting y into one of the original equations. Let's use the first equation:

7x - 2(3) = -6

Simplify the equation:

7x - 6 = -6

Add 6 to both sides:

7x = 0

Divide both sides by 7:

x = 0

Conclusion

In this article, we solved a system of linear equations using the substitution method. We found that the values of x and y that satisfy both equations are x = 0 and y = 3.

Answer

The correct answer is:

B. (0, 3)

This is the solution to the system of linear equations.

Discussion

Solving a system of linear equations is an essential concept in mathematics and is used extensively in various fields. In this article, we used the substitution method to solve a system of two linear equations with two variables. We found that the values of x and y that satisfy both equations are x = 0 and y = 3.

References

Algebra: A comprehensive textbook on algebra that covers systems of linear equations.
Linear Algebra: A textbook on linear algebra that covers systems of linear equations and other related topics.
Mathematics: A textbook on mathematics that covers systems of linear equations and other related topics.

Glossary

System of Linear Equations: A set of two or more linear equations that are solved simultaneously to find the values of the variables.
Substitution Method: A method for solving a system of linear equations that involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination Method: A method for solving a system of linear equations that involves adding or subtracting the equations to eliminate one variable.
Graphical Method: A method for solving a system of linear equations that involves graphing the equations on a coordinate plane and finding the point of intersection.

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Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: What are the different methods for solving a system of linear equations?

A: There are several methods for solving a system of linear equations, including:

Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation.
Elimination Method: This method involves adding or subtracting the equations to eliminate one variable.
Graphical Method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: How do I choose the best method for solving a system of linear equations?

A: The best method for solving a system of linear equations depends on the specific equations and the variables involved. If the equations are simple and easy to solve, the substitution method may be the best choice. If the equations are more complex, the elimination method may be more effective. If the equations are graphed on a coordinate plane, the graphical method may be the best choice.

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

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

Not checking for extraneous solutions: Make sure to check for extraneous solutions, which are solutions that do not satisfy both equations.
Not using the correct method: Choose the best method for solving the system of linear equations.
Not simplifying the equations: Simplify the equations before solving them to make it easier to find the solution.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, substitute the solution into both equations and make sure it satisfies both equations.

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

A: Solving systems of linear equations is an essential concept in mathematics and is used extensively in various fields, including physics, engineering, economics, and computer science.

Q: How do I apply the concepts of solving systems of linear equations in real-life situations?

A: The concepts of solving systems of linear equations can be applied in various real-life situations, such as:

Physics: Solving systems of linear equations is used to describe the motion of objects and to calculate forces and energies.
Engineering: Solving systems of linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations is used to model economic systems and to make predictions about economic trends.
Computer Science: Solving systems of linear equations is used in computer graphics and game development to create realistic simulations and animations.

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

A: Some common applications of solving systems of linear equations include:

Computer Graphics: Solving systems of linear equations is used to create realistic simulations and animations.
Game Development: Solving systems of linear equations is used to create realistic game physics and simulations.
Optimization: Solving systems of linear equations is used to optimize systems, such as electrical circuits and mechanical systems.
Data Analysis: Solving systems of linear equations is used to analyze and model data in various fields, including economics and finance.

Q: How do I practice solving systems of linear equations?

A: To practice solving systems of linear equations, try the following:

Practice problems: Practice solving systems of linear equations with different types of equations and variables.
Real-life applications: Apply the concepts of solving systems of linear equations to real-life situations.
Online resources: Use online resources, such as video tutorials and practice problems, to help you practice solving systems of linear equations.

Q: What are some resources for learning more about solving systems of linear equations?

A: Some resources for learning more about solving systems of linear equations include:

Textbooks: Textbooks on algebra and linear algebra that cover systems of linear equations.
Online resources: Online resources, such as video tutorials and practice problems, that cover systems of linear equations.
Tutorials: Tutorials and workshops that cover systems of linear equations.
Practice problems: Practice problems and exercises that cover systems of linear equations.