Solve The Equation.$\frac{2}{3} - 4x + \frac{7}{2} = -9x + \frac{5}{6}$

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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, 23−4x+72=−9x+56\frac{2}{3} - 4x + \frac{7}{2} = -9x + \frac{5}{6}, using a step-by-step approach. We will break down the solution into manageable parts, making it easier to understand and follow.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its components. The equation is a linear equation, which means it can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants. In this case, the equation is:

23−4x+72=−9x+56\frac{2}{3} - 4x + \frac{7}{2} = -9x + \frac{5}{6}

Our goal is to isolate the variable xx and find its value.

Step 1: Simplify the Equation

To simplify the equation, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 33, 22, and 66 is 66. So, we multiply both sides of the equation by 66:

6(23−4x+72)=6(−9x+56)6 \left( \frac{2}{3} - 4x + \frac{7}{2} \right) = 6 \left( -9x + \frac{5}{6} \right)

This simplifies to:

4−24x+21=−54x+54 - 24x + 21 = -54x + 5

Step 2: Combine Like Terms

Now that we have simplified the equation, we can combine like terms. We can combine the constants on the left-hand side and the constants on the right-hand side:

25−24x=−54x+525 - 24x = -54x + 5

Step 3: Isolate the Variable

Our goal is to isolate the variable xx. We can do this by getting all the terms with xx on one side of the equation and the constants on the other side. We can start by adding 54x54x to both sides of the equation:

25+30x=525 + 30x = 5

Step 4: Solve for x

Now that we have isolated the variable xx, we can solve for its value. We can do this by subtracting 2525 from both sides of the equation:

30x=−2030x = -20

Next, we can divide both sides of the equation by 3030:

x=−2030x = -\frac{20}{30}

Simplifying the fraction, we get:

x=−23x = -\frac{2}{3}

Conclusion

Solving linear equations is an essential skill in mathematics, and it requires a step-by-step approach. In this article, we solved the equation 23−4x+72=−9x+56\frac{2}{3} - 4x + \frac{7}{2} = -9x + \frac{5}{6} using a systematic approach. We simplified the equation, combined like terms, isolated the variable, and finally solved for its value. By following these steps, we can solve linear equations with ease.

Tips and Tricks

  • Always simplify the equation before solving it.
  • Combine like terms to make the equation easier to work with.
  • Isolate the variable by getting all the terms with the variable on one side of the equation.
  • Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

Practice Problems

  • Solve the equation 2x+5=3x−22x + 5 = 3x - 2.
  • Solve the equation x−3=2x+1x - 3 = 2x + 1.
  • Solve the equation 4x+2=2x−34x + 2 = 2x - 3.

Introduction

In our previous article, we solved the equation 23−4x+72=−9x+56\frac{2}{3} - 4x + \frac{7}{2} = -9x + \frac{5}{6} using a step-by-step approach. In this article, we will answer some frequently asked questions about solving linear equations. Whether you are a student or a professional, this Q&A guide will help you understand the concepts and techniques involved in solving linear equations.

Q: What is a linear equation?

A linear equation is an equation in which the highest power of the variable (usually x) is 1. It can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants.

Q: How do I simplify a linear equation?

To simplify a linear equation, you need to get rid of the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

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

The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

Q: How do I combine like terms in a linear equation?

To combine like terms in a linear equation, you need to add or subtract the coefficients of the terms with the same variable.

Q: What is the difference between a linear equation and a quadratic equation?

A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a linear equation with fractions?

To solve a linear equation with fractions, you need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to get rid of the fractions.

Q: What is the order of operations in solving a linear equation?

The order of operations in solving a linear equation is:

  1. Simplify the equation by getting rid of fractions.
  2. Combine like terms.
  3. Isolate the variable by getting all the terms with the variable on one side of the equation.
  4. Solve for the variable.

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

Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your work by solving the equation manually.

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

Some common mistakes to avoid when solving linear equations include:

  • Not simplifying the equation before solving it.
  • Not combining like terms.
  • Not isolating the variable.
  • Not checking your work.

Conclusion

Solving linear equations is an essential skill in mathematics, and it requires a step-by-step approach. By understanding the concepts and techniques involved in solving linear equations, you can become more confident in your ability to solve them. Whether you are a student or a professional, this Q&A guide will help you understand the basics of solving linear equations.

Practice Problems

  • Solve the equation 2x+5=3x−22x + 5 = 3x - 2.
  • Solve the equation x−3=2x+1x - 3 = 2x + 1.
  • Solve the equation 4x+2=2x−34x + 2 = 2x - 3.

By practicing these problems, you can improve your skills in solving linear equations and become more confident in your ability to solve them.

Additional Resources

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

By using these resources, you can get additional help and practice solving linear equations.