The Given Algebraic Manipulation

Oct 13, 2025 by magicalceylontours.com 34 views

Introduction

Algebraic manipulations are an essential part of mathematics, and understanding the justification behind each step is crucial for solving equations accurately. In this article, we will analyze a given algebraic manipulation and provide a step-by-step explanation of the justification for each step.

The Given Algebraic Manipulation

The given algebraic manipulation is as follows:

{ \begin{aligned} 3x + 9 - 7x & = x + 10 + x \\ -4x + 9 & = 2x + 10 \\ -6x + 9 & = 10 \\ -6x & = 1 \\ x & = -\frac{1}{6} \end{aligned} \}

Step 1: Justification for the Subtraction Property of Equality

The first step involves subtracting $7x$ from both sides of the equation:

$3x + 9 - 7x = x + 10 + x$

The justification for this step is based on the subtraction property of equality, which states that if two expressions are equal, then we can subtract the same value from both expressions without changing their equality. In this case, we are subtracting $7x$ from both sides, which is a valid operation.

Step 2: Justification for the Simplification of the Equation

The next step involves simplifying the equation by combining like terms:

$-4x + 9 = 2x + 10$

The justification for this step is based on the commutative property of addition, which states that the order of the terms does not change the result. In this case, we are rearranging the terms to simplify the equation.

Step 3: Justification for the Isolation of the Variable

The next step involves isolating the variable $x$ by subtracting $2x$ from both sides of the equation:

$-6x + 9 = 10$

The justification for this step is based on the addition property of equality, which states that if two expressions are equal, then we can add the same value to both expressions without changing their equality. In this case, we are subtracting $2x$ from both sides to isolate the variable.

Step 4: Justification for the Isolation of the Variable (continued)

The next step involves isolating the variable $x$ by subtracting $9$ from both sides of the equation:

$-6x = 1$

Step 5: Justification for the Solution of the Equation

The final step involves solving the equation by dividing both sides by $-6$ :

$x = -\frac{1}{6}$

The justification for this step is based on the multiplication property of equality, which states that if two expressions are equal, then we can multiply both expressions by the same non-zero value without changing their equality. In this case, we are dividing both sides by $-6$ to solve the equation.

Conclusion

In conclusion, the justification for each step in the given algebraic manipulation is based on the fundamental properties of equality, including the subtraction property, commutative property, addition property, and multiplication property. By understanding these properties, we can ensure that our algebraic manipulations are accurate and valid.

Common Algebraic Properties

The following are some common algebraic properties that are used in algebraic manipulations:

Addition Property of Equality: If $a = b$ , then $a + c = b + c$ .
Subtraction Property of Equality: If $a = b$ , then $a - c = b - c$ .
Multiplication Property of Equality: If $a = b$ , then $ac = bc$ .
Division Property of Equality: If $a = b$ , then $\frac{a}{c} = \frac{b}{c}$ .
Commutative Property of Addition: $a + b = b + a$ .
Commutative Property of Multiplication: $ab = ba$ .
Associative Property of Addition: $(a + b) + c = a + (b + c)$ .
Associative Property of Multiplication: $(ab)c = a(bc)$ .

Real-World Applications

Algebraic manipulations have numerous real-world applications, including:

Physics: Algebraic manipulations are used to describe the motion of objects and to solve problems involving forces and energies.
Engineering: Algebraic manipulations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Computer Science: Algebraic manipulations are used to develop algorithms and to solve problems involving data structures and graph theory.
Economics: Algebraic manipulations are used to model economic systems and to solve problems involving supply and demand.

Final Thoughts

Q: What is the purpose of algebraic manipulations?

A: The purpose of algebraic manipulations is to simplify complex equations and to solve for unknown variables. Algebraic manipulations involve using various properties of equality, such as the addition property, subtraction property, multiplication property, and division property, to isolate the variable and solve the equation.

Q: What are some common algebraic properties used in algebraic manipulations?

A: Some common algebraic properties used in algebraic manipulations include:

Addition Property of Equality: If $a = b$ , then $a + c = b + c$ .
Subtraction Property of Equality: If $a = b$ , then $a - c = b - c$ .
Multiplication Property of Equality: If $a = b$ , then $ac = bc$ .
Division Property of Equality: If $a = b$ , then $\frac{a}{c} = \frac{b}{c}$ .
Commutative Property of Addition: $a + b = b + a$ .
Commutative Property of Multiplication: $ab = ba$ .
Associative Property of Addition: $(a + b) + c = a + (b + c)$ .
Associative Property of Multiplication: $(ab)c = a(bc)$ .

Q: How do I know which algebraic property to use in a given situation?

A: To determine which algebraic property to use, you need to analyze the equation and identify the operation that needs to be performed. For example, if you need to add or subtract the same value from both sides of the equation, you can use the addition or subtraction property of equality. If you need to multiply or divide both sides of the equation by the same non-zero value, you can use the multiplication or division property of equality.

Q: What are some common mistakes to avoid when performing algebraic manipulations?

A: Some common mistakes to avoid when performing algebraic manipulations include:

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when performing algebraic manipulations.
Not using the correct algebraic property: Make sure to use the correct algebraic property for the given situation.
Not checking for extraneous solutions: Make sure to check for extraneous solutions by plugging the solution back into the original equation.
Not simplifying the equation: Make sure to simplify the equation by combining like terms and eliminating any unnecessary variables.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solution back into the original equation and verify that it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What are some real-world applications of algebraic manipulations?

A: Algebraic manipulations have numerous real-world applications, including:

Physics: Algebraic manipulations are used to describe the motion of objects and to solve problems involving forces and energies.
Engineering: Algebraic manipulations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Computer Science: Algebraic manipulations are used to develop algorithms and to solve problems involving data structures and graph theory.
Economics: Algebraic manipulations are used to model economic systems and to solve problems involving supply and demand.

Q: How can I practice algebraic manipulations?

A: To practice algebraic manipulations, you can try the following:

Work on algebraic manipulations problems: Try working on algebraic manipulations problems from a textbook or online resource.
Use online resources: Use online resources, such as Khan Academy or Mathway, to practice algebraic manipulations.
Join a study group: Join a study group or find a study partner to practice algebraic manipulations together.
Take online courses: Take online courses or watch video tutorials to learn more about algebraic manipulations.

Conclusion

In conclusion, algebraic manipulations are an essential part of mathematics, and understanding the justification behind each step is crucial for solving equations accurately. By understanding the fundamental properties of equality and applying them to algebraic manipulations, we can ensure that our solutions are accurate and valid.