Multiplying the Second Equation

Oct 13, 2025 by miodeya.com 115 views

Introduction

Systems of linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a system of two linear equations with two variables. We will use the method of elimination to solve the system, which involves multiplying one or both equations by a suitable constant to eliminate one of the variables.

The System of Equations

The system of equations we will be solving is:

\begin{array}{l} 6x - 3y = 3 \\ -2x + 6y = 14 \end{array}

Eliminating the x-Terms

To eliminate the x-terms when adding the two equations, we need to multiply the second equation by a suitable constant. The constant we need to multiply the second equation by is the reciprocal of the coefficient of x in the first equation, which is 6. However, we also need to consider the coefficient of x in the second equation, which is -2. To eliminate the x-terms, we need to multiply the second equation by a constant that will make the coefficient of x in the second equation equal to the coefficient of x in the first equation.

Let's analyze the coefficients of x in both equations:

In the first equation, the coefficient of x is 6.
In the second equation, the coefficient of x is -2.

To eliminate the x-terms, we need to multiply the second equation by a constant that will make the coefficient of x in the second equation equal to 6. Since the coefficient of x in the second equation is -2, we need to multiply the second equation by -3, which is the reciprocal of -2.

Multiplying the Second Equation

To eliminate the x-terms, we will multiply the second equation by -3:

-3(-2x + 6y) = -3(14)

Expanding the equation, we get:

6x - 18y = -42

Adding the Two Equations

Now that we have multiplied the second equation by -3, we can add the two equations to eliminate the x-terms:

6x - 3y = 3

6x - 18y = -42

Adding the two equations, we get:

-15y = -39

Solving for y

To solve for y, we can divide both sides of the equation by -15:

y = \frac{-39}{-15}

Simplifying the equation, we get:

y = \frac{13}{5}

Solving for x

Now that we have solved for y, we can substitute the value of y into one of the original equations to solve for x. We will use the first equation:

6x - 3y = 3

Substituting the value of y, we get:

6x - 3\left(\frac{13}{5}\right) = 3

Simplifying the equation, we get:

6x - \frac{39}{5} = 3

Multiplying both sides of the equation by 5, we get:

30x - 39 = 15

Adding 39 to both sides of the equation, we get:

30x = 54

Dividing both sides of the equation by 30, we get:

x = \frac{54}{30}

Simplifying the equation, we get:

x = \frac{9}{5}

Conclusion

In this article, we solved a system of two linear equations with two variables using the method of elimination. We multiplied the second equation by a suitable constant to eliminate the x-terms, and then added the two equations to solve for y. Finally, we substituted the value of y into one of the original equations to solve for x. The solution to the system is x = 9/5 and y = 13/5.

Example Use Cases

Systems of linear equations have many real-world applications, including:

Physics and Engineering: Systems of linear equations are used to model physical systems, such as the motion of objects under the influence of gravity or friction.
Economics: Systems of linear equations are used to model economic systems, such as the supply and demand of goods and services.
Computer Science: Systems of linear equations are used in computer graphics, machine learning, and data analysis.

Tips and Tricks

Use the method of elimination: The method of elimination is a powerful tool for solving systems of linear equations. It involves multiplying one or both equations by a suitable constant to eliminate one of the variables.
Check your work: It's essential to check your work when solving systems of linear equations. Make sure to verify that the solution satisfies both equations.
Use technology: Technology can be a powerful tool for solving systems of linear equations. Use graphing calculators, computer software, or online tools to help you solve systems of linear equations.

Conclusion

Introduction

Solving systems of linear equations is a fundamental concept in mathematics, and it's essential to understand the concepts and techniques involved. In this article, we will provide a Q&A guide to help you better understand the topic.

Q: What is a system of linear equations?

A system of linear equations is a set of two or more linear equations that involve two or more variables. Each equation is a linear equation, which means it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

There are several methods for solving systems 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 of the variables.
Graphing method: This method involves graphing the equations on a coordinate plane and finding the point of intersection.
Matrix method: This method involves using matrices to represent the system of equations and then solving for the variables.

Q: What is the elimination method?

The elimination method is a technique for solving systems of linear equations by adding or subtracting the equations to eliminate one of the variables. This method involves multiplying one or both equations by a suitable constant to eliminate one of the variables.

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

The choice of method depends on the type of system and the variables involved. If the system has two variables and two equations, the substitution or elimination method may be the best choice. If the system has more than two variables or equations, the matrix method may be more suitable.

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

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

Not checking the work: It's essential to verify that the solution satisfies both equations.
Not using the correct method: Choosing the wrong method can lead to incorrect solutions.
Not simplifying the equations: Simplifying the equations can make it easier to solve the system.

Q: How do I check my work when solving a system of linear equations?

To check your work, substitute the solution into both equations and verify that it satisfies both equations. If the solution does not satisfy both equations, you may need to re-evaluate your work.

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

Systems of linear equations have many real-world applications, including:

Physics and Engineering: Systems of linear equations are used to model physical systems, such as the motion of objects under the influence of gravity or friction.
Economics: Systems of linear equations are used to model economic systems, such as the supply and demand of goods and services.
Computer Science: Systems of linear equations are used in computer graphics, machine learning, and data analysis.

Q: How can I use technology to solve systems of linear equations?

There are many software programs and online tools available that can help you solve systems of linear equations, including:

Graphing calculators: Graphing calculators can be used to graph the equations and find the point of intersection.
Computer algebra systems: Computer algebra systems, such as Mathematica or Maple, can be used to solve systems of linear equations.
Online tools: Online tools, such as Wolfram Alpha or Symbolab, can be used to solve systems of linear equations.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics, and it's essential to understand the concepts and techniques involved. In this article, we provided a Q&A guide to help you better understand the topic. We covered the different methods for solving systems of linear equations, common mistakes to avoid, and real-world applications. We also discussed how to use technology to solve systems of linear equations.