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Introduction

Solving systems of equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. 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:

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

Our goal is to find the values of $x$ and $y$ that satisfy both equations.

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. Let's analyze the coefficients of the $x$-terms 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 the coefficient of $x$ in the first equation. In other words, we need to multiply the second equation by a constant that will make the coefficient of $x$ in the second equation equal to 6.

Finding the Multiplication Factor

To find the multiplication factor, we can divide the coefficient of $x$ in the first equation (6) by the coefficient of $x$ in the second equation (-2):

$$\frac{6}{-2} = -3$$

Therefore, we need to multiply the second equation by -3 to eliminate the $x$-terms.

Multiplying the Second Equation

Let's multiply the second equation by -3:

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

Expanding the left-hand side, 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:

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

$$y = \frac{13}{5}$$

Solving for $x$

Now that we have found the value of $y$, we can substitute it into one of the original equations to solve for $x$. Let's use the first equation:

$$6x - 3y = 3$$

Substituting $y = \frac{13}{5}$, we get:

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

Simplifying the left-hand side, we get:

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

Multiplying both sides by 5 to eliminate the fraction, we get:

$$30x - 39 = 15$$

Adding 39 to both sides, we get:

$$30x = 54$$

Dividing both sides by 30, we get:

$$x = \frac{54}{30}$$

$$x = \frac{9}{5}$$

Conclusion

In this article, we have 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 values of $x$ and $y$ that satisfy both equations are $x = \frac{9}{5}$ and $y = \frac{13}{5}$.

Example Use Cases

Solving systems of equations has many practical applications in various fields, including:

• Physics: Solving systems of equations is essential in physics to describe the motion of objects under the influence of forces.
• Engineering: Solving systems of equations is crucial in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Solving systems of equations is a fundamental concept in computer science, particularly in the field of numerical analysis.

Tips and Tricks

When solving systems of equations, it is essential to:

• Check your work: Always check your work to ensure that the solution satisfies both equations.
• Use the correct method: Choose the correct method to solve the system, such as elimination or substitution.
• Simplify the equations: Simplify the equations before solving them to make the solution process easier.

Introduction

Solving systems of equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. In this article, we will provide a Q&A guide to help you understand the concepts and methods involved in 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 two or more variables. The goal is to find the values of the variables that satisfy all the equations in the system.

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

A: There are two main methods for solving systems of equations:

• Elimination method: This method involves eliminating one of the variables by adding or subtracting the equations.
• Substitution method: This method involves substituting one of the variables into the other equation to solve for the remaining variable.

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

A: To choose the correct method, you need to examine the equations and determine which method is more suitable. If the coefficients of one of the variables are the same in both equations, the elimination method is more suitable. If the coefficients of one of the variables are different in both equations, the substitution method is more suitable.

Q: What is the elimination method?

A: The elimination method involves eliminating one of the variables by adding or subtracting the equations. This is done by multiplying one or both of the equations by a suitable constant to make the coefficients of the variable to be eliminated the same.

Q: What is the substitution method?

A: The substitution method involves substituting one of the variables into the other equation to solve for the remaining variable. This is done by solving one of the equations for one of the variables and then substituting that expression into the other equation.

Q: How do I solve a system of equations using the elimination method?

A: To solve a system of equations using the elimination method, follow these steps:

1. Examine the equations and determine which variable to eliminate.
2. Multiply one or both of the equations by a suitable constant to make the coefficients of the variable to be eliminated the same.
3. Add or subtract the equations to eliminate the variable.
4. Solve for the remaining variable.
5. Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

Q: How do I solve a system of equations using the substitution method?

A: To solve a system of equations using the substitution method, follow these steps:

1. Examine the equations and determine which variable to solve for first.
2. Solve one of the equations for the variable to be solved for.
3. Substitute the expression into the other equation.
4. Solve for the remaining variable.
5. Substitute the value of the remaining variable into one of the original equations to solve for the other variable.

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 work: Always check the work to ensure that the solution satisfies both equations.
• Using the wrong method: Choose the correct method for solving the system.
• Not simplifying the equations: Simplify the equations before solving them to make the solution process easier.
• Not following the steps: Follow the steps carefully to ensure that the solution is correct.

Conclusion

Solving systems of equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. By following the steps outlined in this Q&A guide, you can become proficient in solving systems of equations and apply this knowledge to various fields.

Example Use Cases

Solving systems of equations has many practical applications in various fields, including:

• Physics: Solving systems of equations is essential in physics to describe the motion of objects under the influence of forces.
• Engineering: Solving systems of equations is crucial in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Solving systems of equations is a fundamental concept in computer science, particularly in the field of numerical analysis.

Tips and Tricks

When solving systems of equations, it is essential to:

• Check your work: Always check your work to ensure that the solution satisfies both equations.
• Use the correct method: Choose the correct method for solving the system.
• Simplify the equations: Simplify the equations before solving them to make the solution process easier.
• Follow the steps: Follow the steps carefully to ensure that the solution is correct.

By following these tips and tricks, you can become proficient in solving systems of equations and apply this knowledge to various fields.