Solve The Equation For \[$ K \$\]:$\[ 8(4k - 4) = -5k - 32 \\]

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, 8(4k - 4) = -5k - 32, to find the value of the variable k. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equation

The given equation is 8(4k - 4) = -5k - 32. To solve for k, we need to isolate the variable on one side of the equation. The equation involves parentheses, which we will need to simplify before solving for k.

Step 1: Simplify the Left Side of the Equation

The left side of the equation is 8(4k - 4). To simplify this expression, we need to multiply 8 by each term inside the parentheses.

8(4k - 4) = 8(4k) - 8(4)

Using the distributive property, we can rewrite the expression as:

32k - 32

So, the simplified left side of the equation is 32k - 32.

Step 2: Simplify the Right Side of the Equation

The right side of the equation is -5k - 32. This expression is already simplified, so we can move on to the next step.

Step 3: Set Up the Equation

Now that we have simplified both sides of the equation, we can set up the equation as follows:

32k - 32 = -5k - 32

Step 4: Add 32 to Both Sides of the Equation

To isolate the variable k, we need to get rid of the constant term on the left side of the equation. We can do this by adding 32 to both sides of the equation.

32k - 32 + 32 = -5k - 32 + 32

Simplifying both sides of the equation, we get:

32k = -5k

Step 5: Add 5k to Both Sides of the Equation

To isolate the variable k, we need to get rid of the negative term on the right side of the equation. We can do this by adding 5k to both sides of the equation.

32k + 5k = -5k + 5k

Simplifying both sides of the equation, we get:

37k = 0

Step 6: Divide Both Sides of the Equation by 37

To find the value of k, we need to divide both sides of the equation by 37.

\frac{37k}{37} = \frac{0}{37}

Simplifying both sides of the equation, we get:

k = 0

Conclusion

Introduction

In our previous article, we solved the linear equation 8(4k - 4) = -5k - 32 to find the value of the variable k. In this article, we will provide a Q&A guide to help you understand the solution and answer common questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an expression. The order of operations is:

  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?

A: To simplify an expression, you need to follow the order of operations. Here are the steps:

  1. Evaluate any expressions 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.

Q: What is the distributive property?

A: The distributive property is a rule that allows you to multiply a single term by multiple terms inside parentheses. The distributive property is written as:

a(b + c) = ab + ac

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable on one side of the equation. Here are the steps:

  1. Simplify both sides of the equation.
  2. Add or subtract the same value to both sides of the equation to isolate the variable.
  3. Multiply or divide both sides of the equation by the same value to isolate the variable.
  4. Check your solution by plugging it back into the original equation.

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(s) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable(s) is 2. For example:

Linear equation: 2x + 3 = 5 Quadratic equation: x^2 + 4x + 4 = 0

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, you need to look at the highest power of the variable(s). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the linear equation 8(4k - 4) = -5k - 32 and answer common questions about solving linear equations. We covered topics such as the order of operations, simplifying expressions, the distributive property, solving linear equations, and the difference between linear and quadratic equations. By following these steps and understanding these concepts, you will be able to solve linear equations with ease.