Solve For { X $} . . . { \frac{3}{4}x + 0.5 = 2x - \frac{1}{3} \}

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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 type of linear equation, which involves fractions and decimals. We will use the given equation as an example and walk through the steps to solve for the unknown variable.

The Given Equation

The given equation is:

34x+0.5=2x−13\frac{3}{4}x + 0.5 = 2x - \frac{1}{3}

This equation involves fractions and decimals, and our goal is to isolate the variable xx.

Step 1: Simplify the Equation

To simplify the equation, we need to get rid of the fractions and decimals. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

The LCM of 4 and 3 is 12, so we will multiply both sides of the equation by 12.

# Multiply both sides of the equation by 12
equation = "12 * (3/4)x + 12 * 0.5 = 12 * (2x - 1/3)"

This simplifies the equation to:

9x+6=24x−49x + 6 = 24x - 4

Step 2: Isolate the Variable

Now that we have simplified the equation, we need to isolate the variable xx. We can do this by getting all the terms with xx on one side of the equation and the constant terms on the other side.

We will start by subtracting 9x9x from both sides of the equation.

# Subtract 9x from both sides of the equation
equation = "6 = 15x - 4"

This simplifies the equation to:

6=15x−46 = 15x - 4

Next, we will add 4 to both sides of the equation.

# Add 4 to both sides of the equation
equation = "10 = 15x"

This simplifies the equation to:

10=15x10 = 15x

Step 3: Solve for x

Now that we have isolated the variable xx, we can solve for its value. We can do this by dividing both sides of the equation by 15.

# Divide both sides of the equation by 15
equation = "x = 10/15"

This simplifies the equation to:

x=23x = \frac{2}{3}

Conclusion

Solving linear equations involves a series of steps, including simplifying the equation, isolating the variable, and solving for its value. In this article, we used the given equation as an example and walked through the steps to solve for the unknown variable. By following these steps, you can solve linear equations involving fractions and decimals.

Tips and Tricks

  • When simplifying the equation, make sure to get rid of the fractions and decimals by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
  • When isolating the variable, make sure to get all the terms with the variable on one side of the equation and the constant terms on the other side.
  • When solving for the variable, make sure to divide both sides of the equation by the coefficient of the variable.

Practice Problems

Try solving the following linear equations:

  1. 23x+1=x−12\frac{2}{3}x + 1 = x - \frac{1}{2}
  2. x+14=2x−32x + \frac{1}{4} = 2x - \frac{3}{2}
  3. 34x−1=2x−13\frac{3}{4}x - 1 = 2x - \frac{1}{3}

References

Glossary

  • Linear Equation: An equation in which the highest power of the variable is 1.
  • Variable: A letter or symbol that represents a value that can change.
  • Coefficient: A number that is multiplied by a variable.
  • Least Common Multiple (LCM): The smallest multiple that two or more numbers have in common.
    Solving Linear Equations: A Q&A Guide =====================================

Introduction

In our previous article, we discussed how to solve linear equations involving fractions and decimals. 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 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.

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

A: The steps to solve a linear equation are:

  1. Simplify the equation by getting rid of any fractions or decimals.
  2. Isolate the variable by getting all the terms with the variable on one side of the equation and the constant terms on the other side.
  3. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to get rid of any fractions or decimals. You can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

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

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. For example, the LCM of 4 and 3 is 12.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to get all the terms with the variable on one side of the equation and the constant terms on the other side. You can do this by adding or subtracting the same value to both sides of the equation.

Q: How do I solve for the variable in a linear equation?

A: To solve for the variable in a linear equation, you need to divide both sides of the equation by the coefficient of the variable.

Q: What is the coefficient of a variable?

A: The coefficient of a variable is the number that is multiplied by the variable. For example, in the equation 2x + 3 = 5, the coefficient of x is 2.

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

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

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 before solving for the variable
  • Not isolating the variable before solving for its value
  • Not checking the solution by plugging it back into the original equation

Q: Can I use algebraic properties to solve linear equations?

A: Yes, you can use algebraic properties to solve linear equations. For example, you can use the distributive property to simplify the equation or the commutative property to rearrange the terms.

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

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

  • Physics: to describe the motion of objects
  • Engineering: to design and optimize systems
  • Economics: to model the behavior of markets
  • Computer Science: to solve problems in computer graphics and game development

Conclusion

Solving linear equations is an essential skill in mathematics and has many real-world applications. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear equations and apply them to a wide range of problems.

Practice Problems

Try solving the following linear equations:

  1. 23x+1=x−12\frac{2}{3}x + 1 = x - \frac{1}{2}
  2. x+14=2x−32x + \frac{1}{4} = 2x - \frac{3}{2}
  3. 34x−1=2x−13\frac{3}{4}x - 1 = 2x - \frac{1}{3}

References

Glossary

  • Linear Equation: An equation in which the highest power of the variable is 1.
  • Variable: A letter or symbol that represents a value that can change.
  • Coefficient: A number that is multiplied by a variable.
  • Least Common Multiple (LCM): The smallest multiple that two or more numbers have in common.