Lucia Tried To Solve The System Below:$\[ \begin{cases} 2x + 2y = 14 \\ x - 2y = -2 \end{cases} \\]Lucia's Work:$\[ x - 2y = -2 \\ x = -2 - 2y \\]What Error Did Lucia Make?A. When Lucia Solved The Second Equation For
Introduction
Solving a system of linear equations is a fundamental concept in mathematics, and it is essential to understand the correct procedures to follow. In this article, we will explore a system of linear equations and examine the work of Lucia, a student who attempted to solve the system. We will identify the error in her work and provide a step-by-step guide on how to solve the system correctly.
The System of Linear Equations
The system of linear equations is given as:
{ \begin{cases} 2x + 2y = 14 \\ x - 2y = -2 \end{cases} \}
This system consists of two linear equations with two variables, x and y. Our goal is to find the values of x and y that satisfy both equations.
Lucia's Work
Lucia's work is as follows:
{ x - 2y = -2 \\ x = -2 - 2y \}
At first glance, it appears that Lucia has made a mistake. Let's examine her work more closely.
Error in Lucia's Work
The error in Lucia's work is that she has only solved the second equation for x, but she has not used the first equation to find the value of y. In other words, she has not used the first equation to eliminate one of the variables.
Step-by-Step Solution
To solve the system of linear equations, we need to follow these steps:
Step 1: Multiply the Two Equations by Necessary Multiples
We can multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same.
{ \begin{cases} 2x + 2y = 14 \\ 2(x - 2y) = 2(-2) \end{cases} \}
This simplifies to:
{ \begin{cases} 2x + 2y = 14 \\ 2x - 4y = -4 \end{cases} \}
Step 2: Subtract the Second Equation from the First Equation
Now, we can subtract the second equation from the first equation to eliminate the variable x.
{ (2x + 2y) - (2x - 4y) = 14 - (-4) \}
This simplifies to:
{ 6y = 18 \}
Step 3: Solve for y
Now, we can solve for y by dividing both sides of the equation by 6.
{ y = \frac{18}{6} \}
This simplifies to:
{ y = 3 \}
Step 4: Substitute the Value of y into One of the Original Equations
Now that we have the value of y, we can substitute it into one of the original equations to find the value of x.
Let's use the first equation:
{ 2x + 2y = 14 \}
Substituting y = 3, we get:
{ 2x + 2(3) = 14 \}
This simplifies to:
{ 2x + 6 = 14 \}
Step 5: Solve for x
Now, we can solve for x by subtracting 6 from both sides of the equation and then dividing both sides by 2.
{ 2x = 14 - 6 \}
This simplifies to:
{ 2x = 8 \}
Dividing both sides by 2, we get:
{ x = \frac{8}{2} \}
This simplifies to:
{ x = 4 \}
Conclusion
In conclusion, Lucia made an error in her work by only solving the second equation for x and not using the first equation to find the value of y. By following the steps outlined above, we were able to solve the system of linear equations and find the values of x and y.
Final Answer
The final answer is x = 4 and y = 3.
Discussion
This problem is a great example of how to solve a system of linear equations. It requires the student to follow a series of steps to eliminate one of the variables and then substitute the value of the other variable into one of the original equations to find the value of the first variable.
Common Mistakes
When solving a system of linear equations, it is common to make mistakes such as:
- Not using the correct equation to eliminate one of the variables
- Not substituting the value of the other variable into one of the original equations
- Not following the correct steps to solve for the variables
Tips and Tricks
To avoid making mistakes when solving a system of linear equations, it is essential to:
- Read the problem carefully and understand what is being asked
- Follow the correct steps to solve for the variables
- Check your work to ensure that you have not made any mistakes
Introduction
Solving systems of linear equations is a fundamental concept in mathematics, and it is essential to understand the correct procedures to follow. In this article, we will provide a Q&A section to help you better understand the concept and address any questions or concerns you may have.
Q1: What is a system of linear equations?
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.
Q2: How do I know if I have a system of linear equations?
You have a system of linear equations if you have two or more linear equations with two or more variables.
Q3: What are the steps to solve a system of linear equations?
The steps to solve a system of linear equations are:
- Multiply the two equations by necessary multiples such that the coefficients of y's in both equations are the same.
- Subtract the second equation from the first equation to eliminate the variable x.
- Solve for y.
- Substitute the value of y into one of the original equations to find the value of x.
Q4: What is the difference between a system of linear equations and a linear equation?
A linear equation is a single equation with one or more variables, while a system of linear equations is a set of two or more linear equations that are solved simultaneously.
Q5: Can I use substitution or elimination to solve a system of linear equations?
Yes, you can use either substitution or elimination to solve a system of linear equations. The choice of method depends on the specific problem and the coefficients of the variables.
Q6: What is the purpose of multiplying the two equations by necessary multiples?
Multiplying the two equations by necessary multiples allows you to eliminate one of the variables by subtracting the second equation from the first equation.
Q7: Can I use a graph to solve a system of linear equations?
Yes, you can use a graph to solve a system of linear equations. However, this method is not always accurate and may not work for all types of systems.
Q8: What is the difference between a dependent and independent system of linear equations?
A dependent system of linear equations has an infinite number of solutions, while an independent system has a unique solution.
Q9: Can I use a calculator to solve a system of linear equations?
Yes, you can use a calculator to solve a system of linear equations. However, it is essential to understand the concept and the steps involved in solving the system.
Q10: What are some common mistakes to avoid when solving a system of linear equations?
Some common mistakes to avoid when solving a system of linear equations include:
- Not using the correct equation to eliminate one of the variables
- Not substituting the value of the other variable into one of the original equations
- Not following the correct steps to solve for the variables
Conclusion
Solving systems of linear equations is a fundamental concept in mathematics, and it is essential to understand the correct procedures to follow. By following the steps outlined in this article and avoiding common mistakes, you can ensure that you are solving systems of linear equations correctly.
Final Tips
- Read the problem carefully and understand what is being asked
- Follow the correct steps to solve for the variables
- Check your work to ensure that you have not made any mistakes
- Use a calculator or graphing tool to help you visualize the problem and check your work
By following these tips and tricks, you can become proficient in solving systems of linear equations and apply this knowledge to a wide range of real-world problems.