Solve For { R $} . . . { -4 = \frac{r}{20} - 5 \}

by ADMIN 50 views

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 type of linear equation, which involves isolating the variable r. We will use a step-by-step approach to solve the equation, and provide explanations and examples to help illustrate the concepts.

The Equation

The equation we will be solving is:

−4=r20−5-4 = \frac{r}{20} - 5

This equation involves a variable r, which we need to isolate. To do this, we will use algebraic manipulations to get r by itself on one side of the equation.

Step 1: Add 5 to Both Sides

The first step in solving this equation is to add 5 to both sides. This will help us eliminate the negative term on the right-hand side of the equation.

−4+5=r20−5+5-4 + 5 = \frac{r}{20} - 5 + 5

Simplifying the left-hand side, we get:

1=r201 = \frac{r}{20}

Step 2: Multiply Both Sides by 20

Next, we need to get rid of the fraction on the right-hand side. To do this, we will multiply both sides of the equation by 20.

1×20=r20×201 \times 20 = \frac{r}{20} \times 20

Simplifying the left-hand side, we get:

20=r20 = r

Conclusion

And that's it! We have successfully solved the equation and isolated the variable r. The final answer is:

r=20r = 20

Tips and Tricks

Here are some tips and tricks to help you solve linear equations like this one:

  • Use inverse operations: When solving an equation, use inverse operations to get rid of the terms on the right-hand side. For example, if you have a term with a negative sign, you can add the opposite of that term to both sides to eliminate it.
  • Simplify the equation: As you solve the equation, simplify it by combining like terms and eliminating fractions.
  • Check your work: Once you think you have solved the equation, check your work by plugging the solution back into the original equation.

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics and engineering: Linear equations are used to model the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Economics: Linear equations are used to model the behavior of economic systems, including supply and demand curves.
  • Computer science: Linear equations are used in computer algorithms, including linear programming and optimization techniques.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

  • Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
  • Not using inverse operations: Failing to use inverse operations can lead to incorrect solutions.
  • Not checking your work: Failing to check your work can lead to incorrect solutions.

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 concepts and techniques involved in 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 are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Add or subtract the same value to both sides: To get rid of the constant term on the right-hand side, add or subtract the same value to both sides of the equation.
  2. Multiply or divide both sides by the same value: To get rid of the coefficient of the variable, multiply or divide both sides of the equation by the same value.
  3. Simplify the equation: Combine like terms and eliminate fractions to simplify the equation.
  4. Check your work: Plug the solution back into the original equation to check that it is true.

Q: How do I handle fractions in a linear equation?

A: To handle fractions in a linear equation, multiply both sides of the equation by the denominator of the fraction. This will eliminate the fraction and allow you to solve the 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, 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: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, it is always a good idea to check your work by plugging the solution back into the original equation to ensure that it is true.

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

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

  • Not simplifying the equation: Failing to simplify the equation can lead to incorrect solutions.
  • Not using inverse operations: Failing to use inverse operations can lead to incorrect solutions.
  • Not checking your work: Failing to check your work can lead to incorrect solutions.

Q: How do I apply linear equations to real-world problems?

A: Linear equations can be applied to a wide range of real-world problems, including:

  • Physics and engineering: Linear equations are used to model the motion of objects, the flow of fluids, and the behavior of electrical circuits.
  • Economics: Linear equations are used to model the behavior of economic systems, including supply and demand curves.
  • Computer science: Linear equations are used in computer algorithms, including linear programming and optimization techniques.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear equations and apply them to real-world problems. Remember to use inverse operations, simplify the equation, and check your work to ensure that you get the correct solution. With practice and patience, you can become a master of solving linear equations.