Distribute the negative sign correctly

Oct 13, 2025 by miodeya.com 15 views

Analyzing Lorena's Work: A Step-by-Step Guide to Solving the Equation

Solving equations is a fundamental concept in mathematics, and it requires a clear understanding of algebraic operations and techniques. In this article, we will analyze Lorena's work on solving the equation $5k - 3\left(2k - \frac{2}{3}\right) - 9 = 0$ . We will examine each step of her solution and identify any errors or misconceptions.

Step 1: Distributing the Negative Sign

Lorena's first step is to distribute the negative sign inside the parentheses:

$5k - 3\left(2k - \frac{2}{3}\right) - 9 = 0$

She writes:

$5k - 6k + 2 - 9 = 0$

However, this step is incorrect. When distributing the negative sign, Lorena should have written:

$5k - 3(2k - \frac{2}{3}) - 9 = 0$

$5k - 6k + 2 + 3(\frac{2}{3}) - 9 = 0$

$5k - 6k + 2 + 2 - 9 = 0$

Step 2: Combining Like Terms

Lorena's next step is to combine like terms:

$5k - 6k + 2 + 2 - 9 = 0$

She writes:

$-k - 7 = 0$

This step is correct. Lorena has combined the like terms correctly.

Step 3: Isolating the Variable

Lorena's next step is to isolate the variable:

$-k - 7 = 0$

She writes:

$-k = 7$

This step is incorrect. Lorena should have added 7 to both sides of the equation to isolate the variable:

$-k - 7 + 7 = 0 + 7$

$-k = 7$

$-k + 7 = 7$

$k = -7$

In conclusion, Lorena's work on solving the equation $5k - 3\left(2k - \frac{2}{3}\right) - 9 = 0$ contains several errors. Specifically, her distribution of the negative sign in Step 1 and her isolation of the variable in Step 3 are incorrect. By following the correct steps and techniques, we can arrive at the correct solution: $k = -7$ .

When solving equations, it's essential to follow the correct steps and techniques. Here are some tips to help you solve equations like a pro:

Distribute the negative sign correctly: When distributing the negative sign, make sure to change the sign of each term inside the parentheses.
Combine like terms carefully: Make sure to combine like terms correctly, and don't forget to include any constants.
Isolate the variable correctly: When isolating the variable, make sure to add or subtract the same value to both sides of the equation.
Check your work: Always check your work by plugging in the solution back into the original equation.

By following these tips and techniques, you can become a master of solving equations and tackle even the most challenging problems with confidence.

When solving equations, it's easy to make mistakes. Here are some common mistakes to avoid:

Distributing the negative sign incorrectly: Make sure to change the sign of each term inside the parentheses when distributing the negative sign.
Combining like terms incorrectly: Make sure to combine like terms correctly, and don't forget to include any constants.
Isolating the variable incorrectly: Make sure to add or subtract the same value to both sides of the equation when isolating the variable.
Not checking your work: Always check your work by plugging in the solution back into the original equation.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

In conclusion, solving equations is a fundamental concept in mathematics that requires a clear understanding of algebraic operations and techniques. By following the correct steps and techniques, and avoiding common mistakes, you can become a master of solving equations and tackle even the most challenging problems with confidence.
Q&A: Solving Equations like a Pro

Solving equations is a fundamental concept in mathematics that requires a clear understanding of algebraic operations and techniques. In our previous article, we analyzed Lorena's work on solving the equation $5k - 3\left(2k - \frac{2}{3}\right) - 9 = 0$ and identified several errors. In this article, we will provide a Q&A section to help you understand the concepts and techniques involved in solving equations.

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to simplify the equation by combining like terms and eliminating any parentheses.

Q: How do I distribute the negative sign correctly?

A: When distributing the negative sign, make sure to change the sign of each term inside the parentheses. For example, if you have the equation $-3(2x - 5)$ , you would distribute the negative sign as follows:

$-3(2x - 5) = -6x + 15$

Q: What is the difference between combining like terms and isolating the variable?

A: Combining like terms involves simplifying the equation by combining terms with the same variable and coefficient. Isolating the variable involves getting the variable alone on one side of the equation.

Q: How do I isolate the variable correctly?

A: To isolate the variable, you need to add or subtract the same value to both sides of the equation. For example, if you have the equation $x + 3 = 7$ , you would isolate the variable as follows:

$x + 3 = 7$

$x + 3 - 3 = 7 - 3$

$x = 4$

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

A: Some common mistakes to avoid when solving equations include:

Distributing the negative sign incorrectly
Combining like terms incorrectly
Isolating the variable incorrectly
Not checking your work

Q: How do I check my work when solving equations?

A: To check your work, plug in the solution back into the original equation. If the solution satisfies the equation, then you have solved the equation correctly.

Q: What are some tips for solving equations like a pro?

A: Some tips for solving equations like a pro include:

Simplifying the equation by combining like terms and eliminating any parentheses
Distributing the negative sign correctly
Isolating the variable correctly
Checking your work
Avoiding common mistakes

Here are some practice problems to help you reinforce your understanding of solving equations:

Solve the equation $2x + 5 = 11$
Solve the equation $x - 3 = 7$
Solve the equation $-2x + 4 = 10$
Solve the equation $x + 2 = 9$
Solve the equation $-x + 3 = 5$

$x = 3$
$x = 10$
$x = 8$
$x = 7$
$x = -2$