Solve The Following Equations:1. { \frac{1}{4} X + \frac{5}{3} X - 4 = 2 - \frac{1}{12} X$}$2. ${$3.2 B - 4.7 = 3 B - 3.3$}$3. ${$4.6 Y - Y + 4 = Y - 1.2$}$4. ${ 7.5 C - 2.5 C + 8 = -7\$}

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**Solving Linear Equations: A Step-by-Step Guide** =====================================================

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 explore four different linear equations and provide step-by-step solutions to each one. We will also discuss the importance of linear equations in real-world applications and provide tips for solving them.

Equation 1: Solving for x

The Equation

14x+53x−4=2−112x\frac{1}{4} x + \frac{5}{3} x - 4 = 2 - \frac{1}{12} x

Step 1: Multiply both sides by the least common multiple (LCM) of the denominators

To eliminate the fractions, we need to multiply both sides of the equation by the LCM of the denominators, which is 12.

12(14x+53x−4)=12(2−112x)12(\frac{1}{4} x + \frac{5}{3} x - 4) = 12(2 - \frac{1}{12} x)

Step 2: Distribute the 12 to each term

3x+20x−48=24−x3x + 20x - 48 = 24 - x

Step 3: Combine like terms

23x−48=24−x23x - 48 = 24 - x

Step 4: Add x to both sides

23x+x−48=2423x + x - 48 = 24

Step 5: Combine like terms

24x−48=2424x - 48 = 24

Step 6: Add 48 to both sides

24x=7224x = 72

Step 7: Divide both sides by 24

x=3x = 3

Equation 2: Solving for b

The Equation

3.2b−4.7=3b−3.33.2 b - 4.7 = 3 b - 3.3

Step 1: Add 4.7 to both sides

3.2b=3b+1.43.2 b = 3 b + 1.4

Step 2: Subtract 3b from both sides

0.2b=1.40.2 b = 1.4

Step 3: Divide both sides by 0.2

b=7b = 7

Equation 3: Solving for y

The Equation

4.6y−y+4=y−1.24.6 y - y + 4 = y - 1.2

Step 1: Combine like terms

3.6y+4=y−1.23.6 y + 4 = y - 1.2

Step 2: Subtract y from both sides

2.6y+4=−1.22.6 y + 4 = -1.2

Step 3: Subtract 4 from both sides

2.6y=−5.22.6 y = -5.2

Step 4: Divide both sides by 2.6

y=−2y = -2

Equation 4: Solving for c

The Equation

7.5c−2.5c+8=−77.5 c - 2.5 c + 8 = -7

Step 1: Combine like terms

5c+8=−75 c + 8 = -7

Step 2: Subtract 8 from both sides

5c=−155 c = -15

Step 3: Divide both sides by 5

c=−3c = -3

Conclusion

Solving linear equations is an essential skill for students and professionals alike. By following the step-by-step solutions provided in this article, you can confidently solve linear equations and apply them to real-world problems. Remember to always multiply both sides of the equation by the LCM of the denominators to eliminate fractions, and to combine like terms to simplify the equation. With practice and patience, you can become proficient in solving linear equations and tackle even the most challenging problems.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics and Engineering: Linear equations are used to describe 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 graphics, game development, and machine learning.
  • Data Analysis: Linear equations are used to analyze and interpret data, including regression analysis and forecasting.

Tips for Solving Linear Equations

  • Read the equation carefully: Make sure you understand what the equation is asking for.
  • Use the correct order of operations: Follow the order of operations (PEMDAS) to simplify the equation.
  • Combine like terms: Simplify the equation by combining like terms.
  • Check your solution: Verify that your solution satisfies the original equation.

By following these tips and practicing regularly, you can become proficient in solving linear equations and apply them to real-world problems.