Aimee Thinks The Solution To The System Below Is $(-4,-6$\]. Eric Thinks The Solution Is $(8,2$\]. Who Is Correct? Show How You Know.System Of Equations:1. $2x - 3y = 10$2. $6y = 4x - 20$Steps To Solve:1. Substitute

Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same variables. Solving a system of linear equations requires finding the values of the variables that satisfy all the equations in the system. In this article, we will solve a system of two linear equations and determine who is correct between Aimee and Eric, who have proposed different solutions to the system.

The System of Linear Equations

The system of linear equations is given by:

1. $2x - 3y = 10$
2. $6y = 4x - 20$

Step 1: Substitute the Second Equation into the First Equation

To solve the system, we can use the method of substitution. We will substitute the expression for $y$ from the second equation into the first equation.

First, we need to isolate $y$ in the second equation:

$6y = 4x - 20$

Divide both sides by 6:

$y = \frac{4x - 20}{6}$

Simplify the expression:

$y = \frac{2x - 10}{3}$

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

$2x - 3y = 10$

Substitute $y = \frac{2x - 10}{3}$:

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

Step 2: Simplify the Equation

Simplify the equation by multiplying both sides by 3:

$6x - (2x - 10) = 30$

Expand and simplify:

$6x - 2x + 10 = 30$

Combine like terms:

$4x + 10 = 30$

Subtract 10 from both sides:

$4x = 20$

Divide both sides by 4:

$x = 5$

Step 3: Find the Value of y

Now that we have found the value of $x$, we can substitute it into one of the original equations to find the value of $y$. We will use the second equation:

$6y = 4x - 20$

Substitute $x = 5$:

$6y = 4(5) - 20$

Simplify:

$6y = 20 - 20$

$6y = 0$

Divide both sides by 6:

$y = 0$

Conclusion

We have solved the system of linear equations and found the values of $x$ and $y$. The solution to the system is $(5, 0)$. Therefore, Aimee is correct, and Eric is incorrect.

Discussion

In this article, we have demonstrated the method of substitution to solve a system of linear equations. We have also shown how to simplify equations and isolate variables. The solution to the system is $(5, 0)$, which confirms that Aimee is correct. This approach can be applied to solve more complex systems of linear equations.

Key Takeaways

• A system of linear equations is a set of two or more linear equations that involve the same variables.
• The method of substitution can be used to solve a system of linear equations.
• To solve a system of linear equations, we can substitute the expression for one variable into the other equation.
• Simplifying equations and isolating variables are essential steps in solving a system of linear equations.

References

• Linear Equations
• System of Linear Equations
• Method of Substitution

Further Reading

• Solving Systems of Linear Equations
• Linear Equations and Inequalities
• Systems of Linear Equations
  Frequently Asked Questions (FAQs) 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 involve the same 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: How do I solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including the method of substitution, the method of elimination, and the method of graphing. The method of substitution involves substituting the expression for one variable into the other equation, while the method of elimination involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the method of substitution?

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

Q: What is the method of elimination?

A: The method of elimination involves adding or subtracting the equations to eliminate one of the variables. This is done by multiplying both equations by necessary multiples such that the coefficients of one of the variables are the same in both equations, and then adding or subtracting the equations.

Q: How do I determine which method to use?

A: The choice of method depends on the specific system of equations and the variables involved. If the equations are easy to solve and the variables are simple, the method of substitution may be the best choice. If the equations are more complex or the variables are more difficult to solve, the method of elimination may be more suitable.

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

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

• Not checking the solution to ensure it satisfies both equations
• Not following the order of operations when simplifying equations
• Not using the correct method for the specific system of equations
• Not checking for extraneous solutions

Q: How do I check my solution to ensure it satisfies both equations?

A: To check your solution, substitute the values of the variables into both equations and simplify. If the result is a true statement, then the solution is correct. If the result is a false statement, then the solution is incorrect.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that satisfies one of the equations but not the other. This can occur when the method of substitution or elimination is used, and the solution is not checked carefully.

Q: How do I avoid extraneous solutions?

A: To avoid extraneous solutions, it is essential to check the solution carefully by substituting the values of the variables into both equations and simplifying. This ensures that the solution satisfies both equations and is not an extraneous solution.

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

A: Solving systems of linear equations has numerous real-world applications, including:

• Physics: Solving systems of linear equations is used to model the motion of objects and predict their trajectories.
• Engineering: Solving systems of linear equations is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Solving systems of linear equations is used to model economic systems and make predictions about the behavior of markets.
• Computer Science: Solving systems of linear equations is used in computer graphics and game development to create realistic simulations and animations.

Conclusion

Solving systems of linear equations is a fundamental concept in mathematics and has numerous real-world applications. By understanding the methods of substitution and elimination, and avoiding common mistakes, you can solve systems of linear equations with confidence. Remember to check your solution carefully to ensure it satisfies both equations and is not an extraneous solution.