What Is The Justification For The Step Taken From Line 2 To Line 3?Given:${ \begin{aligned} 3x + 9 - 7x & = X + 10 + X \ -4x + 9 & = 2x + 10 \ -6x + 9 & = 10 \ -6x & = 1 \ x & = -\frac{1}{6} \end{aligned} }$Options:A. The Subtraction
Introduction
In algebra, we often perform a series of steps to simplify an equation and solve for the unknown variable. However, it's essential to understand the justification behind each step taken. In this article, we'll examine a specific equation and analyze the step taken from line 2 to line 3.
The Given Equation
The given equation is:
Analyzing the Step from Line 2 to Line 3
Let's focus on the step taken from line 2 to line 3:
The question is: what is the justification for subtracting 2x from both sides of the equation in line 2?
The Correct Justification
The correct justification for subtracting 2x from both sides of the equation in line 2 is to isolate the variable x on one side of the equation. By subtracting 2x from both sides, we are essentially moving the term 2x to the left-hand side of the equation, which allows us to combine like terms and simplify the equation.
The Incorrect Justification
However, the step taken from line 2 to line 3 is not justified by simply subtracting 2x from both sides of the equation. The correct step would be to subtract 2x from both sides of the equation in line 2, resulting in:
The Correct Solution
The correct solution to the equation is:
Conclusion
In conclusion, the step taken from line 2 to line 3 is not justified by simply subtracting 2x from both sides of the equation. The correct justification is to subtract 2x from both sides of the equation in line 2, resulting in a simplified equation that can be solved for the unknown variable x.
Importance of Justification
Understanding the justification behind each step taken in algebra is crucial for solving equations correctly. By analyzing the steps taken, we can identify any errors or incorrect assumptions and correct them accordingly. This ensures that we arrive at the correct solution to the equation.
Tips for Algebraic Manipulations
When performing algebraic manipulations, it's essential to:
- Understand the rules of algebra: Familiarize yourself with the rules of algebra, including the order of operations, distributive property, and combining like terms.
- Justify each step: Take the time to justify each step taken in the solution, ensuring that you understand the reasoning behind each manipulation.
- Check your work: Double-check your work to ensure that the solution is correct and that each step is justified.
Q: What is the importance of justification in algebra?
A: Justification is crucial in algebra as it ensures that each step taken in the solution is correct and logical. By understanding the justification behind each step, we can identify any errors or incorrect assumptions and correct them accordingly.
Q: How do I determine if a step is justified?
A: To determine if a step is justified, ask yourself:
- Is the step consistent with the rules of algebra?
- Does the step follow logically from the previous step?
- Is the step necessary to simplify the equation or solve for the unknown variable?
Q: What are some common mistakes to avoid when performing algebraic manipulations?
A: Some common mistakes to avoid when performing algebraic manipulations include:
- Incorrectly applying the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
- Failing to combine like terms: Combine like terms to simplify the equation and avoid unnecessary steps.
- Not justifying each step: Take the time to justify each step taken in the solution to ensure that it is correct and logical.
Q: How can I improve my understanding of algebraic manipulations?
A: To improve your understanding of algebraic manipulations, try the following:
- Practice, practice, practice: Practice solving equations and simplifying expressions to develop your skills and build confidence.
- Work through examples: Work through examples and exercises to understand the steps involved in solving equations and simplifying expressions.
- Seek help when needed: Don't be afraid to ask for help if you're struggling with a concept or step.
Q: What are some real-world applications of algebraic manipulations?
A: Algebraic manipulations have numerous real-world applications, including:
- Science and engineering: Algebraic manipulations are used to model and solve problems in physics, chemistry, and engineering.
- Computer science: Algebraic manipulations are used in computer science to develop algorithms and solve problems in computer graphics, cryptography, and more.
- Economics: Algebraic manipulations are used in economics to model and solve problems in finance, economics, and more.
Q: How can I use algebraic manipulations to solve real-world problems?
A: To use algebraic manipulations to solve real-world problems, try the following:
- Identify the problem: Identify the problem you want to solve and determine the variables involved.
- Develop a model: Develop a mathematical model to represent the problem and the variables involved.
- Solve the equation: Use algebraic manipulations to solve the equation and find the solution.
Q: What are some common algebraic manipulations?
A: Some common algebraic manipulations include:
- Simplifying expressions: Simplify expressions by combining like terms and applying the order of operations.
- Solving equations: Solve equations by isolating the variable and using inverse operations.
- Graphing functions: Graph functions by plotting points and using algebraic manipulations to find the equation of the graph.
Q: How can I use algebraic manipulations to graph functions?
A: To use algebraic manipulations to graph functions, try the following:
- Plot points: Plot points on the coordinate plane to represent the function.
- Use algebraic manipulations: Use algebraic manipulations to find the equation of the graph and determine the x- and y-intercepts.
- Graph the function: Graph the function using the points and equation found in the previous steps.