Solve For $w$:$-2(-6w + 7) - W = 5(w - 4) - 2$Simplify Your Answer As Much As Possible. $w =$

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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, which involves simplifying and isolating the variable w. We will break down the solution into manageable steps, making it easier to understand and follow.

The Given Equation

The given equation is:

−2(−6w+7)−w=5(w−4)−2-2(-6w + 7) - w = 5(w - 4) - 2

Our goal is to solve for w, which means we need to isolate the variable w on one side of the equation.

Step 1: Distribute the Negative 2

To simplify the equation, we will start by distributing the negative 2 to the terms inside the parentheses.

−2(−6w+7)=12w−14-2(-6w + 7) = 12w - 14

So, the equation becomes:

12w−14−w=5(w−4)−212w - 14 - w = 5(w - 4) - 2

Step 2: Combine Like Terms

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

12w−w=11w−1412w - w = 11w - 14

The equation now becomes:

11w−14=5(w−4)−211w - 14 = 5(w - 4) - 2

Step 3: Distribute the 5

To simplify the equation further, we will distribute the 5 to the terms inside the parentheses.

5(w−4)=5w−205(w - 4) = 5w - 20

So, the equation becomes:

11w−14=5w−20−211w - 14 = 5w - 20 - 2

Step 4: Combine Like Terms Again

We will combine the like terms on the right-hand side of the equation.

5w−20−2=5w−225w - 20 - 2 = 5w - 22

The equation now becomes:

11w−14=5w−2211w - 14 = 5w - 22

Step 5: Add 14 to Both Sides

To isolate the variable w, we will add 14 to both sides of the equation.

11w−14+14=5w−22+1411w - 14 + 14 = 5w - 22 + 14

This simplifies to:

11w=5w−811w = 5w - 8

Step 6: Subtract 5w from Both Sides

Next, we will subtract 5w from both sides of the equation.

11w−5w=−811w - 5w = -8

This simplifies to:

6w=−86w = -8

Step 7: Divide Both Sides by 6

Finally, we will divide both sides of the equation by 6 to solve for w.

6w6=−86\frac{6w}{6} = \frac{-8}{6}

This simplifies to:

w=−43w = -\frac{4}{3}

Conclusion

Introduction

In our previous article, we solved a linear equation step by step, simplifying and isolating the variable w. In this article, we will answer some frequently asked questions about solving linear equations, providing additional insights and tips to help you master this essential math skill.

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 are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Distribute any negative numbers or coefficients to the terms inside the parentheses.
  2. Combine like terms on both sides of the equation.
  3. Add or subtract the same value to both sides of the equation to isolate the variable.
  4. Multiply or divide both sides of the equation by the same value to solve for the variable.

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, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. In other words, a linear equation can be written in the form ax + b = c, while a quadratic equation can be written in the form ax^2 + bx + c = 0.

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

A: To determine if an equation is linear or quadratic, look for 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.

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

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

  • Not distributing negative numbers or coefficients to the terms inside the parentheses.
  • Not combining like terms on both sides of the equation.
  • Not adding or subtracting the same value to both sides of the equation to isolate the variable.
  • Not multiplying or dividing both sides of the equation by the same value to solve for the variable.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

  • Using online resources, such as math websites or apps.
  • Working with a tutor or teacher to practice solving linear equations.
  • Creating your own practice problems to solve.
  • Using real-world examples to apply linear equations to practical problems.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects, such as the trajectory of a projectile.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits or mechanical systems.
  • Economics: Linear equations are used to model economic systems, such as supply and demand curves.
  • Computer Science: Linear equations are used in algorithms and data structures, such as linear search and sorting algorithms.

Conclusion

In this article, we answered some frequently asked questions about solving linear equations, providing additional insights and tips to help you master this essential math skill. By following the steps outlined in this article, you can become proficient in solving linear equations and apply them to real-world problems.