Solve For $w$.$3(3w + 10) = 120$Simplify Your Answer As Much As Possible.$ W = W = W = [/tex]

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear equation, $3(3w + 10) = 120$, and provide a step-by-step guide on how to simplify the solution.

Understanding the Equation

The given equation is a linear equation in one variable, $w$. The equation is in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants. In this case, $a = 3$, $b = 10$, and $c = 120$. Our goal is to isolate the variable $w$ and find its value.

Step 1: Distribute the Coefficient

The first step in solving the equation is to distribute the coefficient $3$ to the terms inside the parentheses. This means multiplying $3$ by each term inside the parentheses.

3(3w+10)=3×3w+3×103(3w + 10) = 3 \times 3w + 3 \times 10

Using the distributive property, we can simplify the equation to:

9w+30=1209w + 30 = 120

Step 2: Subtract 30 from Both Sides

The next step is to isolate the term with the variable $w$. To do this, we need to subtract $30$ from both sides of the equation.

9w+30−30=120−309w + 30 - 30 = 120 - 30

Simplifying the equation, we get:

9w=909w = 90

Step 3: Divide Both Sides by 9

Now that we have isolated the term with the variable $w$, we can divide both sides of the equation by $9$ to find the value of $w$.

9w9=909\frac{9w}{9} = \frac{90}{9}

Simplifying the equation, we get:

w=10w = 10

Conclusion

In this article, we solved a linear equation, $3(3w + 10) = 120$, using a step-by-step approach. We distributed the coefficient, subtracted $30$ from both sides, and finally divided both sides by $9$ to find the value of $w$. The final solution is $w = 10$.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS).
  • Make sure to distribute the coefficient to all terms inside the parentheses.
  • When subtracting or adding the same value to both sides of the equation, be careful not to change the equation's balance.
  • When dividing both sides of the equation by a value, make sure to check if the value is zero to avoid division by zero errors.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Practice Problems

Try solving the following linear equations:

  1. 2(x+5)=162(x + 5) = 16

  2. 4(y−3)=204(y - 3) = 20

  3. 3(z+2)=153(z + 2) = 15

Solutions

  1. x=3x = 3

  2. y=7y = 7

  3. z=3z = 3

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations using a step-by-step approach. In this article, we will provide a Q&A guide to help you better understand the concept of 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. It is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Distribute the coefficient to all terms inside the parentheses.
  2. Combine like terms.
  3. Add or subtract the same value to both sides of the equation.
  4. Divide both sides of the equation by the coefficient.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to distribute a coefficient to all terms inside the parentheses. It states that a(b + c) = ab + ac.

Q: How do I know if an equation is linear or not?

A: To determine if an equation is linear or not, look for the highest power of the variable(s). If the highest power is 1, then the equation is linear.

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 the coefficient to all terms inside the parentheses.
  • Not combining like terms.
  • Adding or subtracting the same value to only one side of the equation.
  • Dividing both sides of the equation by zero.

Q: How do I check if my solution is correct?

A: To check if your solution is correct, plug the value back into the original equation and simplify. If the equation is true, then your solution is correct.

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

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

  • Physics: Linear equations are used to describe the motion of objects under constant acceleration.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about future trends.

Q: How do I practice solving linear equations?

A: To practice solving linear equations, try solving the following problems:

  1. 2(x+5)=162(x + 5) = 16

  2. 4(y−3)=204(y - 3) = 20

  3. 3(z+2)=153(z + 2) = 15

Q: What are some advanced topics related to linear equations?

A: Some advanced topics related to linear equations include:

  • Systems of linear equations
  • Linear inequalities
  • Matrix operations

Conclusion

Solving linear equations is a fundamental skill that has numerous real-world applications. By following a step-by-step approach and using the distributive property, we can solve linear equations and find the value of the variable. Remember to avoid common mistakes and check your solution to ensure accuracy. With practice, you'll become proficient in solving linear equations and be able to apply them to real-world problems.

Practice Problems

Try solving the following linear equations:

  1. 5(x−2)=255(x - 2) = 25

  2. 3(y+4)=213(y + 4) = 21

  3. 2(z−1)=82(z - 1) = 8

Solutions

  1. x=7x = 7

  2. y=5y = 5

  3. z=5z = 5

Additional Resources

For more information on solving linear equations, check out the following resources:

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram Alpha: Linear Equations