Which Ordered Pair Below Is A Solution To The Following System Of Equations?${ \begin{array}{l} 4x + 3y = 2 \ 3x - 2y = 10 \end{array} }$A. { (-2, 2)$}$ B. { (-2, -2)$}$ C. { (2, -2)$}$ D. [$(2,

Introduction

When dealing with a system of linear equations, it's essential to find the ordered pair that satisfies both equations simultaneously. In this article, we will explore how to solve a system of linear equations using the substitution method and the elimination method. We will also apply these methods to the given system of equations to determine the correct ordered pair.

Understanding the System of Equations

The given system of equations is:

{ \begin{array}{l} 4x + 3y = 2 \\ 3x - 2y = 10 \end{array} \}

Our goal is to find the ordered pair $(x, y)$ that satisfies both equations.

The Substitution Method

One way to solve a system of linear equations is by using the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation.

Let's start by solving the first equation for $y$:

$4x + 3y = 2$

$3y = -4x + 2$

$y = \frac{-4x + 2}{3}$

Now, substitute this expression for $y$ into the second equation:

$3x - 2y = 10$

$3x - 2\left(\frac{-4x + 2}{3}\right) = 10$

$3x + \frac{8x - 4}{3} = 10$

$9x + 8x - 4 = 30$

$17x = 34$

$x = 2$

Now that we have found the value of $x$, we can substitute it back into the expression for $y$:

$y = \frac{-4x + 2}{3}$

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

$y = \frac{-8 + 2}{3}$

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

$y = -2$

Therefore, the ordered pair $(x, y)$ that satisfies both equations is $(2, -2)$.

The Elimination Method

Another way to solve a system of linear equations is by using the elimination method. This method involves adding or subtracting the equations to eliminate one variable.

Let's start by multiplying the first equation by 2 and the second equation by 3:

$8x + 6y = 4$

$9x - 6y = 30$

Now, add both equations to eliminate the variable $y$:

$17x = 34$

$x = 2$

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

$4x + 3y = 2$

$4(2) + 3y = 2$

$8 + 3y = 2$

$3y = -6$

$y = -2$

Therefore, the ordered pair $(x, y)$ that satisfies both equations is $(2, -2)$.

Conclusion

In this article, we have explored how to solve a system of linear equations using the substitution method and the elimination method. We have applied these methods to the given system of equations and determined that the correct ordered pair is $(2, -2)$.

Discussion

Which method do you prefer when solving a system of linear equations? Do you have any questions or concerns about the substitution method or the elimination method? Share your thoughts and experiences in the comments below.

Final Answer

The final answer is: $\boxed{(2, -2)}$

Introduction

Solving systems of linear equations can be a challenging task, but with the right approach and techniques, it can be made easier. In this article, we will address some of the most frequently asked questions about solving systems of linear equations.

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 two main methods for solving systems of linear equations?

A: The two main methods for solving systems of linear equations are the substitution method and the elimination method.

Q: What is the substitution method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do I choose which method to use?

A: The choice of method depends on the form of the equations and the variables involved. If the equations are in the form of $y = mx + b$, the substitution method may be easier to use. If the equations have the same coefficient for one variable, the elimination method may be easier to use.

Q: What if I have a system of three or more equations?

A: If you have a system of three or more equations, you can use the substitution method or the elimination method to solve for two variables, and then substitute those values into the remaining equations to solve for the third variable.

Q: What if I have a system of equations with fractions or decimals?

A: If you have a system of equations with fractions or decimals, you can multiply both sides of the equation by a common multiple to eliminate the fractions or decimals.

Q: Can I use a calculator to solve systems of linear equations?

A: Yes, you can use a calculator to solve systems of linear equations. Many calculators have built-in functions for solving systems of linear equations.

Q: What if I get stuck or make a mistake?

A: If you get stuck or make a mistake, don't worry! Take a step back and review the equations and the steps you've taken. You can also try using a different method or seeking help from a teacher or tutor.

Q: Are there any shortcuts or tricks for solving systems of linear equations?

A: Yes, there are several shortcuts and tricks for solving systems of linear equations. For example, you can use the method of substitution to solve for one variable, and then use the method of elimination to solve for the other variable.

Conclusion

Solving systems of linear equations can be a challenging task, but with the right approach and techniques, it can be made easier. By understanding the two main methods for solving systems of linear equations and practicing with different types of equations, you can become proficient in solving systems of linear equations.

Additional Resources

If you're looking for additional resources to help you with solving systems of linear equations, here are a few suggestions:

• Online tutorials and videos
• Practice problems and worksheets
• Calculator software and apps
• Online communities and forums

Final Tips

• Practice, practice, practice! The more you practice solving systems of linear equations, the more comfortable you'll become with the different methods and techniques.
• Don't be afraid to ask for help if you're stuck or make a mistake.
• Use a calculator or other tools to check your work and ensure accuracy.

Final Answer

The final answer is: There is no single final answer, as the solution to a system of linear equations depends on the specific equations and variables involved.