Solve For $w$.$\frac{w-3}{10}=\frac{3}{4}$Simplify Your Answer.$w =$

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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 type of linear equation, where the variable is isolated on one side of the equation. We will use the given equation w−310=34\frac{w-3}{10}=\frac{3}{4} as an example and walk through the step-by-step process of solving for ww.

Understanding the Equation

The given equation is a linear equation in the form of a fraction. The equation is w−310=34\frac{w-3}{10}=\frac{3}{4}. Our goal is to isolate the variable ww on one side of the equation.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 10 and 4 is 20.

\frac{w-3}{10}=\frac{3}{4}
\implies 20 \times \frac{w-3}{10} = 20 \times \frac{3}{4}
\implies 2(w-3) = 15

Step 2: Distribute the Multiplication

Now that we have multiplied both sides by 20, we can distribute the multiplication to simplify the equation.

2(w-3) = 15
\implies 2w - 6 = 15

Step 3: Add 6 to Both Sides

To isolate the variable ww, we need to get rid of the constant term on the same side as the variable. We can do this by adding 6 to both sides of the equation.

2w - 6 = 15
\implies 2w - 6 + 6 = 15 + 6
\implies 2w = 21

Step 4: Divide Both Sides by 2

Finally, we can isolate the variable ww by dividing both sides of the equation by 2.

2w = 21
\implies \frac{2w}{2} = \frac{21}{2}
\implies w = \frac{21}{2}

Conclusion

In this article, we have walked through the step-by-step process of solving a linear equation with a fraction. We have used the given equation w−310=34\frac{w-3}{10}=\frac{3}{4} as an example and have isolated the variable ww on one side of the equation. The final solution is w=212w = \frac{21}{2}.

Tips and Tricks

  • When solving linear equations with fractions, it is essential to multiply both sides by the least common multiple (LCM) of the denominators.
  • Distribute the multiplication to simplify the equation.
  • Add or subtract the same value to both sides of the equation to isolate the variable.
  • Divide both sides of the equation by the coefficient of the variable to isolate the variable.

Common Mistakes to Avoid

  • Failing to multiply both sides by the LCM of the denominators.
  • Not distributing the multiplication to simplify the equation.
  • Adding or subtracting the wrong value to both sides of the equation.
  • Dividing both sides of the equation by the wrong coefficient.

Real-World Applications

Solving linear equations with fractions has numerous real-world applications, including:

  • Finance: Calculating interest rates and investment returns.
  • Science: Measuring physical quantities and rates of change.
  • Engineering: Designing and optimizing systems and structures.

Conclusion

Introduction

In our previous article, we walked through the step-by-step process of solving a linear equation with a fraction. We have isolated the variable ww on one side of the equation and have arrived at the final solution w=212w = \frac{21}{2}. However, we understand that solving linear equations with fractions can be a challenging task, and students often have questions and doubts. In this article, we will address some of the most frequently asked questions and provide additional tips and tricks to help students master this skill.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 10 and 4 is 20.

Q: Why do I need to multiply both sides of the equation by the LCM?

A: Multiplying both sides of the equation by the LCM eliminates the fractions and allows us to work with whole numbers. This makes it easier to isolate the variable and solve the equation.

Q: What if the LCM is not a whole number?

A: If the LCM is not a whole number, we can still multiply both sides of the equation by the LCM. However, we need to be careful when simplifying the equation to avoid introducing fractions.

Q: Can I use a different method to solve the equation?

A: Yes, there are other methods to solve linear equations with fractions, such as using algebraic manipulations or graphing. However, the method we used in our previous article is a straightforward and efficient way to solve the equation.

Q: What if I get stuck or make a mistake?

A: Don't worry! Making mistakes is a natural part of the learning process. If you get stuck or make a mistake, take a step back and review the equation. Check your work and make sure you are following the correct steps.

Q: How can I practice solving linear equations with fractions?

A: There are many online resources and practice exercises available to help you practice solving linear equations with fractions. You can also try solving equations with different denominators and coefficients to challenge yourself.

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

A: Solving linear equations with fractions has numerous real-world applications, including:

  • Finance: Calculating interest rates and investment returns.
  • Science: Measuring physical quantities and rates of change.
  • Engineering: Designing and optimizing systems and structures.

Q: Can I use a calculator to solve linear equations with fractions?

A: Yes, you can use a calculator to solve linear equations with fractions. However, it's essential to understand the underlying math and be able to solve the equation by hand.

Conclusion

In conclusion, solving linear equations with fractions is a crucial skill for students to master. By following the step-by-step process outlined in our previous article and addressing common questions and doubts, students can confidently solve equations with fractions and apply their knowledge to real-world problems.

Additional Tips and Tricks

  • Practice, practice, practice! The more you practice solving linear equations with fractions, the more comfortable you will become with the process.
  • Use online resources and practice exercises to supplement your learning.
  • Challenge yourself by solving equations with different denominators and coefficients.
  • Review and practice algebraic manipulations and graphing to develop a deeper understanding of linear equations.

Common Mistakes to Avoid

  • Failing to multiply both sides by the LCM.
  • Not distributing the multiplication to simplify the equation.
  • Adding or subtracting the wrong value to both sides of the equation.
  • Dividing both sides of the equation by the wrong coefficient.

Real-World Applications

Solving linear equations with fractions has numerous real-world applications, including:

  • Finance: Calculating interest rates and investment returns.
  • Science: Measuring physical quantities and rates of change.
  • Engineering: Designing and optimizing systems and structures.

Conclusion

In conclusion, solving linear equations with fractions is a crucial skill for students to master. By following the step-by-step process outlined in our previous article and addressing common questions and doubts, students can confidently solve equations with fractions and apply their knowledge to real-world problems.