What Is The Solution To The System Of Equations?$\[ \begin{cases} 10x + 3y = 43 \\ -9x - Y = -20 \end{cases} \\]A. (11, 1) B. (1, 11) C. (-1, 11) D. (11, -1)

Introduction to Systems of Equations

A system of equations is a set of two or more equations that contain multiple variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations with two variables.

The System of Equations

The given system of equations is:

$\begin{cases} 10x + 3y = 43 \\ -9x - y = -20 \end{cases}$

Step 1: Multiply the Two Equations by Necessary Multiples

To eliminate one of the variables, we can multiply the two equations by necessary multiples such that the coefficients of either x or y in both equations are the same.

Let's multiply the first equation by 1 and the second equation by 3.

$\begin{cases} 10x + 3y = 43 \\ -27x - 3y = -60 \end{cases}$

Step 2: Add the Two Equations

Now, we can add the two equations to eliminate the variable y.

$(10x + 3y) + (-27x - 3y) = 43 + (-60)$

This simplifies to:

$-17x = -17$

Step 3: Solve for x

To solve for x, we can divide both sides of the equation by -17.

$x = \frac{-17}{-17}$

$x = 1$

Step 4: Substitute x into One of the Original Equations

Now that we have the value of x, we can substitute it into one of the original equations to solve for y.

Let's substitute x into the first equation:

$10(1) + 3y = 43$

This simplifies to:

$10 + 3y = 43$

Step 5: Solve for y

To solve for y, we can subtract 10 from both sides of the equation and then divide both sides by 3.

$3y = 43 - 10$

$3y = 33$

$y = \frac{33}{3}$

$y = 11$

Conclusion

The solution to the system of equations is x = 1 and y = 11.

Answer

The correct answer is B. (1, 11).

Discussion

Solving a system of equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we used the method of elimination to solve a system of two linear equations with two variables. We multiplied the two equations by necessary multiples, added the two equations to eliminate one of the variables, and then solved for the other variable. This method can be used to solve systems of equations with two or more variables.

Tips and Tricks

• When solving a system of equations, it's essential to identify the variables and the constants in each equation.
• The method of elimination involves multiplying the equations by necessary multiples and then adding or subtracting the equations to eliminate one of the variables.
• When solving for a variable, make sure to isolate the variable on one side of the equation.
• Check your solution by substituting the values of the variables into each equation to ensure that they satisfy all the equations in the system.

Related Topics

• Solving systems of equations using substitution
• Solving systems of equations using matrices
• Solving systems of nonlinear equations
• Graphing systems of equations

References

• [1] "Systems of Equations" by Math Open Reference
• [2] "Solving Systems of Equations" by Khan Academy
• [3] "Systems of Linear Equations" by MIT OpenCourseWare

Note: The references provided are for informational purposes only and are not a substitute for the original sources.

Introduction

Solving systems of equations can be a challenging task, especially for those who are new to the concept. In this article, we will address some of the most frequently asked questions 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 contain multiple variables. Solving a system of equations involves finding 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 several methods for solving systems of equations, including:

• Substitution method: This involves solving one equation for one variable and then substituting that expression 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 difference between a linear equation and a nonlinear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. For example, 2x + 3y = 5 is a linear equation. A nonlinear equation, on the other hand, is an equation in which the highest power of the variable(s) is greater than 1. For example, x^2 + 3y = 5 is a nonlinear equation.

Q: How do I know which method to use?

A: The choice of method depends on the type of system of equations you are dealing with. If the system is linear and has two variables, the substitution or elimination method may be the best choice. If the system is nonlinear or has more than two variables, the matrix method may be more suitable.

Q: What is the importance of checking the solution?

A: It is essential to check the solution by substituting the values of the variables into each equation to ensure that they satisfy all the equations in the system. This helps to avoid errors and ensures that the solution is correct.

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

A: Yes, you can use a calculator to solve systems of equations. Many calculators have built-in functions for solving systems of equations, including the substitution and elimination methods.

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: Failing to check the solution can lead to incorrect answers.
• Not following the order of operations: Failing to follow the order of operations can lead to incorrect answers.
• Not using the correct method: Using the wrong method can lead to incorrect answers.

Q: How can I practice solving systems of equations?

A: You can practice solving systems of equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of equations on your own using a calculator or a computer program.

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 describe the motion of objects in physics.
• Engineering: Solving systems of equations is used to design and optimize systems in engineering.
• Economics: Solving systems of equations is used to model economic systems and make predictions about economic trends.

Conclusion

Solving systems of equations is a fundamental concept in mathematics that has many real-world applications. By understanding the different methods for solving systems of equations and practicing solving systems of equations, you can become proficient in this area and apply it to a variety of fields.