What Is The Solution To The System Of Equations?${ \begin{array}{l} 2x + 4y = 12 \ y = \frac{1}{4}x - 3 \end{array} }$A. { (-1, 8)$}$B. { (8, -1)$}$C. { \left(5, \frac{1}{2}\right)$}$D.

Mar 10, 2025 by ADMIN 187 views

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

Introduction

Solving a system of equations is a fundamental concept in mathematics that involves finding the values of variables that satisfy multiple equations simultaneously. In this article, we will explore a system of two linear equations and provide a step-by-step solution to find the values of the variables.

The System of Equations

The given system of equations is:

{ \begin{array}{l} 2x + 4y = 12 \\ y = \frac{1}{4}x - 3 \end{array} \}

Understanding the Equations

The first equation is a linear equation in two variables, x and y. It can be written in the form Ax + By = C, where A, B, and C are constants. In this case, A = 2, B = 4, and C = 12.

The second equation is also a linear equation in two variables, x and y. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept. In this case, m = 1/4 and b = -3.

Substitution Method

To solve the system of equations, we can use the substitution method. This involves substituting the expression for y from the second equation into the first equation.

Step 1: Substitute the Expression for y

Substitute the expression for y from the second equation into the first equation:

2x + 4y = 12

y = (1/4)x - 3

Substituting the expression for y into the first equation, we get:

2x + 4((1/4)x - 3) = 12

Step 2: Simplify the Equation

Simplify the equation by combining like terms:

2x + x - 12 = 12

Combine like terms:

3x - 12 = 12

Step 3: Add 12 to Both Sides

Add 12 to both sides of the equation to isolate the term with x:

3x = 24

Step 4: Divide Both Sides by 3

Divide both sides of the equation by 3 to solve for x:

x = 8

Step 5: Substitute the Value of x into the Second Equation

Substitute the value of x into the second equation to solve for y:

y = (1/4)x - 3

y = (1/4)(8) - 3

y = 2 - 3

y = -1

The Solution

The solution to the system of equations is x = 8 and y = -1.

Conclusion

In this article, we have explored a system of two linear equations and provided a step-by-step solution to find the values of the variables. The substitution method was used to solve the system of equations, and the solution was found to be x = 8 and y = -1.

Answer

The correct answer is:

B. (8, -1)

Discussion

This problem is a classic example of a system of linear equations. The substitution method was used to solve the system of equations, and the solution was found to be x = 8 and y = -1. This problem can be used to illustrate the concept of solving a system of equations and can be used as a teaching tool for students.

References

[1] "Solving Systems of Linear Equations" by Math Open Reference
[2] "Solving Systems of Linear Equations" by Khan Academy
[3] "Solving Systems of Linear Equations" by Wolfram MathWorld

Keywords

System of equations
Linear equations
Substitution method
Elimination method
Graphing
Nonlinear equations
Frequently Asked Questions (FAQs) About Solving Systems of Equations ====================================================================

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that involve the same variables. In other words, it is a collection of equations that are related to each other through the variables.

Q: What are the different methods for solving systems of equations?

A: There are several methods for solving systems of equations, including:

Substitution method: This involves substituting the expression for one variable from one equation into the other equation.
Elimination method: This involves adding or subtracting the equations to eliminate one of the variables.
Graphing method: This involves graphing the equations on a coordinate plane and finding the point of intersection.
Matrix method: This involves using matrices to solve the system of equations.

Q: What is the substitution method?

A: The substitution method is a method for solving systems of equations that involves substituting the expression for one variable from one equation into the other equation. This method is useful when one of the equations is already solved for one of the variables.

Q: What is the elimination method?

A: The elimination method is a method for solving systems of equations that involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of the variables in the two equations are the same.

Q: What is the graphing method?

A: The graphing method is a method for solving systems of equations that involves graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the system of equations is linear.

Q: What is the matrix method?

A: The matrix method is a method for solving systems of equations that involves using matrices to solve the system of equations. This method is useful when the system of equations is large and complex.

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

A: The best method for solving a system of equations depends on the type of equations and the variables involved. If the equations are linear and the variables are easy to solve for, the substitution or elimination method may be the best choice. If the equations are nonlinear or the variables are difficult to solve for, the graphing or matrix method may be the best choice.

Q: What are some common mistakes to avoid when solving systems of equations?

A: Some common mistakes to avoid when solving systems of equations include:

Not checking the solution to make sure it satisfies both equations.
Not using the correct method for the type of equations.
Not simplifying the equations before solving.
Not checking for extraneous solutions.

Q: How do I check my solution to make sure it is correct?

A: To check your solution, plug the values of the variables back into both equations and make sure they are true. If the solution satisfies both equations, it is correct. If it does not satisfy both equations, it is incorrect.

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

A: Solving systems of equations has many real-world applications, including:

Physics: Solving systems of equations is used to model the motion of objects and to solve problems involving forces and energies.
Engineering: Solving systems of equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of equations is used to model economic systems and to solve problems involving supply and demand.
Computer Science: Solving systems of equations is used in computer graphics and game development to create realistic simulations and animations.

Q: How can I practice solving systems of equations?

A: There are many ways to practice solving systems of equations, including:

Using online resources, such as Khan Academy and Mathway.
Working with a tutor or teacher.
Practicing with worksheets and problems.
Using real-world examples and applications.

Q: What are some common types of systems of equations?

A: Some common types of systems of equations include:

Linear systems: These are systems of equations where the variables are linear.
Nonlinear systems: These are systems of equations where the variables are nonlinear.
Homogeneous systems: These are systems of equations where the constant term is zero.
Inhomogeneous systems: These are systems of equations where the constant term is not zero.

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

A: Some common techniques for solving systems of equations include:

Substitution method: This involves substituting the expression for one variable from one equation into the other equation.
Elimination method: This involves adding or subtracting the equations to eliminate one of the variables.
Graphing method: This involves graphing the equations on a coordinate plane and finding the point of intersection.
Matrix method: This involves using matrices to solve the system of equations.