Simplify The Expression:${ 8(x-3)-(6-2x) = 2(x+2)-5(57x) }$

Introduction

Algebraic equations are a fundamental concept in mathematics, and simplifying expressions is a crucial skill to master. In this article, we will focus on simplifying a given expression using algebraic techniques. The expression we will be working with is:

$$8(x-3)-(6-2x) = 2(x+2)-5(57x)$$

This equation involves various operations, including multiplication, subtraction, and addition. Our goal is to simplify the expression by combining like terms and performing the necessary operations.

Distributive Property and Order of Operations

Before we begin simplifying the expression, it's essential to understand the distributive property and the order of operations. The distributive property states that for any real numbers a, b, and c:

$$a(b+c) = ab + ac$$

This property allows us to distribute a single term to multiple terms inside parentheses. The order of operations, on the other hand, is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the Expression

Now that we have a solid understanding of the distributive property and the order of operations, let's begin simplifying the expression.

First, we will apply the distributive property to the left-hand side of the equation:

$$8(x-3) = 8x - 24$$

$$-(6-2x) = -6 + 2x$$

So, the left-hand side of the equation becomes:

$$8x - 24 - 6 + 2x$$

Next, we will combine like terms on the left-hand side of the equation:

$$8x + 2x = 10x$$

$$-24 - 6 = -30$$

Therefore, the left-hand side of the equation simplifies to:

$$10x - 30$$

Now, let's simplify the right-hand side of the equation:

$$2(x+2) = 2x + 4$$

$$-5(57x) = -285x$$

So, the right-hand side of the equation becomes:

$$2x + 4 - 285x$$

Next, we will combine like terms on the right-hand side of the equation:

$$2x - 285x = -283x$$

$$4 - 285x$$

Therefore, the right-hand side of the equation simplifies to:

$$-283x + 4$$

Combining Like Terms

Now that we have simplified both sides of the equation, we can combine like terms to further simplify the expression:

$$10x - 30 = -283x + 4$$

We can combine the x-terms by adding or subtracting their coefficients:

$$10x + 283x = 293x$$

So, the equation becomes:

$$293x - 30 = 4$$

Solving for x

Now that we have simplified the expression, we can solve for x by isolating the variable on one side of the equation. To do this, we will add 30 to both sides of the equation:

$$293x - 30 + 30 = 4 + 30$$

This simplifies to:

$$293x = 34$$

Next, we will divide both sides of the equation by 293 to solve for x:

$$\frac{293x}{293} = \frac{34}{293}$$

This simplifies to:

$$x = \frac{34}{293}$$

Conclusion

In this article, we simplified a given expression using algebraic techniques. We applied the distributive property and the order of operations to simplify the expression, and then combined like terms to further simplify the expression. Finally, we solved for x by isolating the variable on one side of the equation. The simplified expression is:

$$x = \frac{34}{293}$$

This result demonstrates the importance of simplifying expressions in algebra and the need to carefully apply mathematical operations to arrive at the correct solution.

Frequently Asked Questions

• Q: What is the distributive property? A: The distributive property is a mathematical property that allows us to distribute a single term to multiple terms inside parentheses.
• Q: What is the order of operations? A: The order of operations is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS.
• Q: How do I simplify an expression using algebraic techniques? A: To simplify an expression using algebraic techniques, apply the distributive property and the order of operations, and then combine like terms to further simplify the expression.

Final Thoughts

Simplifying expressions is a crucial skill in algebra, and it requires careful attention to detail and a solid understanding of mathematical operations. By applying the distributive property and the order of operations, and combining like terms, we can simplify complex expressions and arrive at the correct solution. Whether you're a student or a professional, mastering the art of simplifying expressions will serve you well in your mathematical endeavors.

Introduction

In our previous article, we simplified a given expression using algebraic techniques. We applied the distributive property and the order of operations to simplify the expression, and then combined like terms to further simplify the expression. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to distribute a single term to multiple terms inside parentheses. For example, if we have the expression:

$$a(b+c)$$

We can apply the distributive property to get:

$$ab + ac$$

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression using algebraic techniques?

A: To simplify an expression using algebraic techniques, follow these steps:

1. Apply the distributive property to any terms inside parentheses.
2. Evaluate any exponential expressions.
3. Evaluate any multiplication and division operations from left to right.
4. Evaluate any addition and subtraction operations from left to right.
5. Combine like terms to further simplify the expression.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. For example, x is a variable. A constant, on the other hand, is a value that does not change. For example, 5 is a constant.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the terms. For example, if we have the expression:

$$2x + 3x$$

We can combine the like terms by adding the coefficients:

$$2x + 3x = 5x$$

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 2x + 1 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

1. Simplify the equation by combining like terms.
2. Isolate the variable on one side of the equation.
3. Solve for the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, follow these steps:

1. Simplify the equation by combining like terms.
2. Factor the equation, if possible.
3. Use the quadratic formula to solve for the variable.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions. We discussed the distributive property, the order of operations, and how to combine like terms. We also discussed the difference between a variable and a constant, and how to solve linear and quadratic equations. By mastering these concepts, you will be able to simplify complex expressions and solve equations with ease.

Frequently Asked Questions

• Q: What is the distributive property? A: The distributive property is a mathematical property that allows us to distribute a single term to multiple terms inside parentheses.
• Q: What is the order of operations? A: The order of operations is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS.
• Q: How do I simplify an expression using algebraic techniques? A: To simplify an expression using algebraic techniques, apply the distributive property and the order of operations, and then combine like terms to further simplify the expression.

Final Thoughts

Simplifying expressions is a crucial skill in algebra, and it requires careful attention to detail and a solid understanding of mathematical operations. By mastering the concepts discussed in this article, you will be able to simplify complex expressions and solve equations with ease. Whether you're a student or a professional, the skills you learn in this article will serve you well in your mathematical endeavors.