Solve The Equation: ${ \frac{2}{3}(4x - 2) = \frac{2}{9}(3x + 3) }$

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Introduction

In this article, we will delve into the world of algebra and solve a complex equation step by step. The equation we will be solving is 23(4x−2)=29(3x+3)\frac{2}{3}(4x - 2) = \frac{2}{9}(3x + 3). This equation involves fractions, variables, and constants, making it a challenging problem for those who are new to algebra. However, with a clear understanding of the steps involved, anyone can solve this equation and gain a deeper understanding of algebraic manipulations.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at what it represents. The equation is a linear equation, which means it can be represented in the form of ax+b=cax + b = c, where aa, bb, and cc are constants. In this case, the equation is 23(4x−2)=29(3x+3)\frac{2}{3}(4x - 2) = \frac{2}{9}(3x + 3). Our goal is to isolate the variable xx and find its value.

Step 1: Distribute the Fractions

To solve the equation, we need to start by distributing the fractions on both sides of the equation. This means we need to multiply each term inside the parentheses by the fraction outside the parentheses. On the left-hand side, we have 23(4x−2)\frac{2}{3}(4x - 2), and on the right-hand side, we have 29(3x+3)\frac{2}{9}(3x + 3).

from fractions import Fraction

frac1 = Fraction(2, 3)
frac2 = Fraction(2, 9)



term1 = 4x - 2
term2 = 3x + 3



left_hand_side = frac1 * term1
right_hand_side = frac2 * term2

Step 2: Simplify the Equation

After distributing the fractions, we need to simplify the equation by combining like terms. On the left-hand side, we have 83x−43\frac{8}{3}x - \frac{4}{3}, and on the right-hand side, we have 69x+69\frac{6}{9}x + \frac{6}{9}.

from fractions import Fraction


frac1 = Fraction(8, 3)
frac2 = Fraction(4, 3)
frac3 = Fraction(6, 9)
frac4 = Fraction(6, 9)



left_hand_side = frac1 * x - frac2
right_hand_side = frac3 * x + frac4

Step 3: Eliminate the Fractions

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 3 and 9 is 9.

from fractions import Fraction


lcm = 9



left_hand_side = lcm * (frac1 * x - frac2)
right_hand_side = lcm * (frac3 * x + frac4)

Step 4: Combine Like Terms

After eliminating the fractions, we need to combine like terms on both sides of the equation. On the left-hand side, we have 24x−1224x - 12, and on the right-hand side, we have 6x+66x + 6.

from fractions import Fraction


term1 = 24x - 12
term2 = 6x + 6



left_hand_side = term1
right_hand_side = term2

Step 5: Isolate the Variable

Finally, we need to isolate the variable xx by moving all the terms containing xx to one side of the equation and the constant terms to the other side. In this case, we can subtract 6x6x from both sides of the equation to get 18x−12=618x - 12 = 6.

from fractions import Fraction


equation = 18*x - 12 - 6



x = (6 + 12) / 18

Conclusion

In this article, we solved the equation 23(4x−2)=29(3x+3)\frac{2}{3}(4x - 2) = \frac{2}{9}(3x + 3) step by step. We distributed the fractions, simplified the equation, eliminated the fractions, combined like terms, and finally isolated the variable xx. By following these steps, anyone can solve this equation and gain a deeper understanding of algebraic manipulations.

Final Answer

The final answer is 33\boxed{\frac{3}{3}}.

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Q&A: Frequently Asked Questions

Q: What is the equation we are solving?

A: The equation we are solving is 23(4x−2)=29(3x+3)\frac{2}{3}(4x - 2) = \frac{2}{9}(3x + 3).

Q: Why do we need to distribute the fractions?

A: We need to distribute the fractions to simplify the equation and make it easier to solve.

Q: What is the least common multiple (LCM) of 3 and 9?

A: The LCM of 3 and 9 is 9.

Q: Why do we need to eliminate the fractions?

A: We need to eliminate the fractions to make it easier to combine like terms and isolate the variable.

Q: What is the final answer to the equation?

A: The final answer to the equation is 33\boxed{\frac{3}{3}}.

Q: Can you explain the steps in more detail?

A: Of course! Let's break down the steps in more detail:

Step 1: Distribute the Fractions

To distribute the fractions, we need to multiply each term inside the parentheses by the fraction outside the parentheses. On the left-hand side, we have 23(4x−2)\frac{2}{3}(4x - 2), and on the right-hand side, we have 29(3x+3)\frac{2}{9}(3x + 3).

from fractions import Fraction


frac1 = Fraction(2, 3)
frac2 = Fraction(2, 9)



term1 = 4x - 2
term2 = 3x + 3



left_hand_side = frac1 * term1
right_hand_side = frac2 * term2

Step 2: Simplify the Equation

After distributing the fractions, we need to simplify the equation by combining like terms. On the left-hand side, we have 83x−43\frac{8}{3}x - \frac{4}{3}, and on the right-hand side, we have 69x+69\frac{6}{9}x + \frac{6}{9}.

from fractions import Fraction


frac1 = Fraction(8, 3)
frac2 = Fraction(4, 3)
frac3 = Fraction(6, 9)
frac4 = Fraction(6, 9)



left_hand_side = frac1 * x - frac2
right_hand_side = frac3 * x + frac4

Step 3: Eliminate the Fractions

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 3 and 9 is 9.

from fractions import Fraction


lcm = 9



left_hand_side = lcm * (frac1 * x - frac2)
right_hand_side = lcm * (frac3 * x + frac4)

Step 4: Combine Like Terms

After eliminating the fractions, we need to combine like terms on both sides of the equation. On the left-hand side, we have 24x−1224x - 12, and on the right-hand side, we have 6x+66x + 6.

from fractions import Fraction


term1 = 24x - 12
term2 = 6x + 6



left_hand_side = term1
right_hand_side = term2

Step 5: Isolate the Variable

Finally, we need to isolate the variable xx by moving all the terms containing xx to one side of the equation and the constant terms to the other side. In this case, we can subtract 6x6x from both sides of the equation to get 18x−12=618x - 12 = 6.

from fractions import Fraction


equation = 18*x - 12 - 6



x = (6 + 12) / 18

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

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

  • Not distributing the fractions correctly
  • Not simplifying the equation properly
  • Not eliminating the fractions correctly
  • Not combining like terms correctly
  • Not isolating the variable correctly

Q: How can I practice solving equations?

A: You can practice solving equations by working through example problems and exercises. You can also try solving equations on your own and then checking your answers with a calculator or a teacher.

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

A: Solving equations has many real-world applications, including:

  • Physics: Solving equations is used to describe the motion of objects and to calculate their velocities and accelerations.
  • Engineering: Solving equations is used to design and optimize systems, such as bridges and buildings.
  • Economics: Solving equations is used to model economic systems and to make predictions about future economic trends.
  • Computer Science: Solving equations is used to develop algorithms and to solve problems in computer science.

Q: Can you provide more examples of equations to solve?

A: Of course! Here are a few more examples of equations to solve:

  • 34(2x+1)=12(x−2)\frac{3}{4}(2x + 1) = \frac{1}{2}(x - 2)
  • 23(x+2)=34(x−1)\frac{2}{3}(x + 2) = \frac{3}{4}(x - 1)
  • 12(3x−2)=23(x+1)\frac{1}{2}(3x - 2) = \frac{2}{3}(x + 1)

I hope these examples help you to practice solving equations and to understand the concepts better.