Solve for x in the second equation

Oct 13, 2025 by windailys.com 125 views

Introduction

In mathematics, solving systems of equations is a fundamental concept that requires careful attention to detail. Lucia, a student, has attempted to solve a system of equations, but unfortunately, she has made an error. In this article, we will analyze her work and identify the mistake she made. By understanding the error, we can provide guidance on how to correctly solve the system of equations.

The System of Equations

The given system of equations is:

{ \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 has attempted to solve the system of equations, but she has made a mistake. Let's examine her work:

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Analyzing Lucia's Work on the System of Equations =====================================================

Q&A: Identifying Lucia's Error

Q: What is the system of equations that Lucia attempted to solve? A: The system of equations is:

{ \begin{cases} 2x + 2y = 14 \\ x - 2y = -2 \end{cases} \}

Q: What was Lucia's first step in solving the system of equations? A: Lucia's first step was to solve for x in the second equation.

Q: What was Lucia's mistake in solving for x? A: Lucia's mistake was that she did not isolate the variable x correctly. She wrote:

{ x - 2y = -2 \}

{ x = -2 + 2y \}

However, she did not simplify the expression further.

Q: How can we simplify the expression for x? A: We can simplify the expression for x by combining like terms:

{ x = -2 + 2y \}

{ x = 2y - 2 \}

Q: What is the next step in solving the system of equations? A: The next step is to substitute the expression for x into the first equation.

Q: How do we substitute the expression for x into the first equation? A: We substitute the expression for x into the first equation by replacing x with 2y - 2:

{ 2(2y - 2) + 2y = 14 \}

Q: How do we simplify the equation after substitution? A: We simplify the equation by distributing the 2 to the terms inside the parentheses:

{ 4y - 4 + 2y = 14 \}

Q: How do we combine like terms in the equation? A: We combine like terms by adding the coefficients of the y terms:

{ 6y - 4 = 14 \}

Q: How do we isolate the variable y in the equation? A: We isolate the variable y by adding 4 to both sides of the equation:

{ 6y = 18 \}

Q: How do we solve for y in the equation? A: We solve for y by dividing both sides of the equation by 6:

{ y = 3 \}

Q: What is the value of x in the system of equations? A: We can find the value of x by substituting the value of y into the expression for x:

{ x = 2y - 2 \}

{ x = 2(3) - 2 \}

{ x = 6 - 2 \}

{ x = 4 \}

Conclusion

In conclusion, Lucia's mistake was that she did not isolate the variable x correctly. By simplifying the expression for x and substituting it into the first equation, we were able to solve for y and then find the value of x. This example illustrates the importance of careful attention to detail when solving systems of equations.

Common Mistakes to Avoid

Not isolating the variable correctly: Make sure to isolate the variable correctly by combining like terms and simplifying the expression.
Not substituting the expression correctly: Make sure to substitute the expression correctly by replacing the variable with the expression.
Not simplifying the equation: Make sure to simplify the equation by combining like terms and eliminating any unnecessary terms.

Tips for Solving Systems of Equations

Read the problem carefully: Make sure to read the problem carefully and understand what is being asked.
Choose the correct method: Choose the correct method for solving the system of equations, such as substitution or elimination.
Simplify the equation: Simplify the equation by combining like terms and eliminating any unnecessary terms.
Check your work: Check your work by plugging the values back into the original equations to make sure they are true.