This Question Has Two Parts. First, Answer Part A. Then, Answer Part B.Part AConsider The Following System Of Equations:${ \begin{array}{l} 6x + 6y = 12 \ 6x - 5y = 12 \end{array} }$Which Arithmetic Operation Would Be Best Used To Solve

Part A: Understanding the System of Equations

When dealing with systems of equations, it's essential to understand the type of equations we're working with. In this case, we have two linear equations with two variables, x and y. The equations are:

{ \begin{array}{l} 6x + 6y = 12 \\ 6x - 5y = 12 \end{array} \}

To solve this system of equations, we need to find the values of x and y that satisfy both equations simultaneously.

Key Concepts:

• Linear Equations: Equations in which the highest power of the variable(s) is 1.
• System of Equations: A set of two or more equations that involve the same variables.
• Solution: The values of the variables that satisfy all the equations in the system.

Step 1: Identify the Type of System

The given system of equations is a linear system with two equations and two variables. We can use various methods to solve this system, including substitution, elimination, and graphing.

Step 2: Choose the Best Method

In this case, the elimination method is the most suitable approach. We can eliminate one of the variables by adding or subtracting the two equations.

Step 3: Apply the Elimination Method

To eliminate the variable x, we can add the two equations:

{ \begin{array}{l} (6x + 6y) + (6x - 5y) = 12 + 12 \\ 12x + y = 24 \end{array} \}

Now, we have a new equation with only one variable, y. We can solve for y by isolating it on one side of the equation.

Step 4: Solve for y

To solve for y, we can subtract 12x from both sides of the equation:

$${ y = 24 - 12x }$$

Now, we have the value of y in terms of x.

Step 5: Substitute y into One of the Original Equations

We can substitute the value of y into one of the original equations to solve for x. Let's use the first equation:

$${ 6x + 6y = 12 }$$

Substituting y = 24 - 12x into this equation, we get:

$${ 6x + 6(24 - 12x) = 12 }$$

Simplifying this equation, we get:

$${ 6x + 144 - 72x = 12 }$$

Combine like terms:

$${ -66x + 144 = 12 }$$

Subtract 144 from both sides:

$${ -66x = -132 }$$

Divide both sides by -66:

$${ x = 2 }$$

Step 6: Find the Value of y

Now that we have the value of x, we can find the value of y by substituting x into one of the original equations. Let's use the first equation:

$${ 6x + 6y = 12 }$$

Substituting x = 2 into this equation, we get:

$${ 6(2) + 6y = 12 }$$

Simplifying this equation, we get:

$${ 12 + 6y = 12 }$$

Subtract 12 from both sides:

$${ 6y = 0 }$$

Divide both sides by 6:

$${ y = 0 }$$

Conclusion

In this article, we solved a system of linear equations using the elimination method. We identified the type of system, chose the best method, applied the elimination method, solved for y, substituted y into one of the original equations, and found the value of x and y.

Part B: Arithmetic Operations for Solving Systems of Equations

When solving systems of equations, we often use arithmetic operations such as addition, subtraction, multiplication, and division. In this section, we'll discuss the best arithmetic operation to use when solving systems of equations.

Addition and Subtraction

Addition and subtraction are useful when we want to eliminate one of the variables. By adding or subtracting the two equations, we can eliminate the variable and solve for the other variable.

Multiplication and Division

Multiplication and division are useful when we want to isolate one of the variables. By multiplying or dividing both sides of the equation by a constant, we can isolate the variable and solve for its value.

Conclusion

In this article, we discussed the best arithmetic operation to use when solving systems of equations. We identified the type of system, chose the best method, applied the elimination method, solved for y, substituted y into one of the original equations, and found the value of x and y. We also discussed the use of addition, subtraction, multiplication, and division when solving systems of equations.

Final Answer

The final answer is:

x = 2 y = 0

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's a collection of equations that need to be solved simultaneously.

Q: What are the different types of systems of equations?

A: There are several types of systems of equations, including:

• Linear systems: These are systems of equations in which the highest power of the variable(s) is 1.
• Non-linear systems: These are systems of equations in which the highest power of the variable(s) is greater than 1.
• Homogeneous systems: These are systems of equations in which the constant term is zero.
• Non-homogeneous systems: These are systems of equations in which the constant term is not zero.

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 method involves substituting one equation into another to solve for one variable.
• Elimination method: This method involves adding or subtracting equations to eliminate one variable.
• Graphing method: This method involves graphing the equations on a coordinate plane to find the point of intersection.
• Matrix method: This method involves using matrices to solve systems of equations.

Q: What is the elimination method?

A: The elimination method is a method for solving systems of equations in which we add or subtract equations to eliminate one variable. This method is useful when the coefficients of the variables are the same in both equations.

Q: What is the substitution method?

A: The substitution method is a method for solving systems of equations in which we substitute one equation into another to solve for one variable. This method is useful when we have one equation with one variable and another equation with both variables.

Q: What is the graphing method?

A: The graphing method is a method for solving systems of equations in which we graph the equations on a coordinate plane to find the point of intersection. This method is useful when we have two linear equations.

Q: What is the matrix method?

A: The matrix method is a method for solving systems of equations in which we use matrices to solve the system. This method is useful when we have a large system of equations.

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

A: To choose the best method for solving a system of equations, you should consider the following factors:

• Type of system: If the system is linear, the elimination method or substitution method may be the best choice. If the system is non-linear, the graphing method or matrix method may be the best choice.
• Number of equations: If the system has two equations, the elimination method or substitution method may be the best choice. If the system has more than two equations, the matrix method may be the best choice.
• Difficulty of the system: If the system is easy to solve, the elimination method or substitution method may be the best choice. If the system is difficult to solve, the graphing method 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: Make sure to check the solution to ensure that it satisfies both equations.
• Not using the correct method: Choose the best method for solving the system based on the type of system and the number of equations.
• Not following the steps: Make sure to follow the steps of the method carefully to avoid errors.
• Not checking for extraneous solutions: Make sure to check for extraneous solutions, which are solutions that do not satisfy one or both of the equations.

Q: How do I check the solution to a system of equations?

A: To check the solution to a system of equations, you should substitute the solution into both equations and check if it satisfies both equations. If the solution satisfies both equations, it is the correct solution. If the solution does not satisfy one or both of the equations, it is an extraneous solution.