Solving Linear Equations $-\frac{2}{9}y + 1 = -3$ A Step-by-Step Guide

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In the realm of mathematics, solving linear equations stands as a fundamental skill, acting as a cornerstone for more advanced concepts. Mastering this skill unlocks the ability to tackle a wide array of mathematical problems, making it an essential tool in various fields. This comprehensive guide delves into the process of solving the linear equation −29y+1=−3-\frac{2}{9}y + 1 = -3, providing a step-by-step approach that empowers you to confidently tackle similar challenges.

Understanding Linear Equations

Before we embark on the solution, let's first grasp the essence of linear equations. A linear equation is essentially a mathematical statement that equates two expressions, where the variable involved is raised to the power of one. These equations can be visualized as straight lines on a graph, hence the term "linear." The equation −29y+1=−3-\frac{2}{9}y + 1 = -3 perfectly fits this definition, featuring the variable 'y' raised to the power of one.

Step-by-Step Solution

Now, let's embark on the journey of solving the equation −29y+1=−3-\frac{2}{9}y + 1 = -3. We'll break down the process into manageable steps, ensuring clarity and understanding.

Step 1: Isolating the Term with the Variable

Our initial goal is to isolate the term containing the variable 'y'. In this case, the term is −29y-\frac{2}{9}y. To achieve this isolation, we need to eliminate the constant term '+1' from the left side of the equation. We can accomplish this by subtracting 1 from both sides of the equation. This maintains the balance of the equation, ensuring that the equality remains valid.

−29y+1−1=−3−1-\frac{2}{9}y + 1 - 1 = -3 - 1

This simplification leads us to:

−29y=−4-\frac{2}{9}y = -4

Step 2: Eliminating the Fractional Coefficient

The next hurdle is the fractional coefficient −29-\frac{2}{9} attached to the variable 'y'. To eliminate this fraction, we can multiply both sides of the equation by the reciprocal of the fraction. The reciprocal of −29-\frac{2}{9} is −92-\frac{9}{2}. Multiplying both sides by this reciprocal will effectively isolate 'y'.

−92∗(−29y)=−4∗(−92)-\frac{9}{2} * (-\frac{2}{9}y) = -4 * (-\frac{9}{2})

This operation simplifies to:

y = 18

Step 3: Checking the Solution

To ensure the accuracy of our solution, it's crucial to verify it. We can do this by substituting the value of 'y' we obtained (y = 18) back into the original equation. If the equation holds true, our solution is correct.

Substituting y = 18 into the original equation −29y+1=−3-\frac{2}{9}y + 1 = -3, we get:

−29∗18+1=−3-\frac{2}{9} * 18 + 1 = -3

Simplifying the left side, we have:

-4 + 1 = -3

-3 = -3

Since the equation holds true, our solution y = 18 is indeed correct.

The Solution Set

The solution set represents the collection of all values that satisfy the equation. In this case, we have found a single value, y = 18, that satisfies the equation. Therefore, the solution set is {18}.

Key Concepts in Solving Linear Equations

To further solidify your understanding, let's highlight some key concepts involved in solving linear equations:

  • The Golden Rule of Equations: The fundamental principle in solving equations is maintaining balance. Any operation performed on one side of the equation must also be performed on the other side to preserve equality.
  • Inverse Operations: Inverse operations are operations that undo each other. For instance, addition and subtraction are inverse operations, as are multiplication and division. We use inverse operations to isolate the variable.
  • Reciprocal: The reciprocal of a fraction is obtained by swapping the numerator and denominator. Multiplying a fraction by its reciprocal results in 1, which is crucial for eliminating fractional coefficients.
  • Solution Set: The solution set is the set of all values that satisfy the equation. It can contain a single value, multiple values, or even be an empty set (indicating no solution).

Applications of Linear Equations

Linear equations are not confined to the realm of mathematics textbooks. They find widespread applications in various real-world scenarios, including:

  • Physics: Linear equations are used to describe motion, forces, and other physical phenomena.
  • Engineering: Engineers utilize linear equations in designing structures, circuits, and systems.
  • Economics: Linear equations play a vital role in modeling supply and demand, cost analysis, and financial planning.
  • Computer Science: Linear equations are used in algorithms, data analysis, and computer graphics.

Practice Problems

To reinforce your understanding and skills, let's tackle some practice problems:

  1. Solve the equation 3x - 5 = 7 and check your solution.
  2. Solve the equation −14z+2=−1-\frac{1}{4}z + 2 = -1 and check your solution.
  3. Solve the equation 2(y + 3) = 10 and check your solution.

Conclusion

Solving linear equations is a fundamental skill with far-reaching applications. By understanding the underlying concepts and following a systematic approach, you can confidently tackle these equations. This guide has equipped you with the knowledge and tools to solve the equation −29y+1=−3-\frac{2}{9}y + 1 = -3 and similar problems. Remember to practice regularly to hone your skills and master this essential mathematical concept.

Solve the equation −29y+1=−3-\frac{2}{9}y + 1 = -3 and check the solution. What is the solution set?