Solving The Equation (12y-1)/2 = (9y+8)/5 A Step-by-step Guide

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In this article, we will delve into the process of finding the value of y that satisfies the equation (12y - 1)/2 = (9y + 8)/5. This is a classic algebraic problem that requires us to manipulate the equation to isolate the variable y. Understanding how to solve such equations is fundamental in mathematics and has applications in various fields.

Understanding the Equation

Before we jump into the solution, let's break down the equation. We have a linear equation with fractions on both sides. The goal is to eliminate the fractions and then isolate y on one side of the equation. This involves using the principles of equality, which state that we can perform the same operation on both sides of an equation without changing its validity. The key steps will involve multiplying both sides by the least common multiple of the denominators to eliminate the fractions, distributing terms, combining like terms, and finally, isolating y.

Step-by-Step Solution

To solve the equation (12y - 1)/2 = (9y + 8)/5, we will follow these steps:

1. Eliminate the Fractions

The first step is 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, which are 2 and 5. The LCM of 2 and 5 is 10. So, we multiply both sides by 10:

10 * [(12y - 1)/2] = 10 * [(9y + 8)/5]

2. Simplify the Equation

Now, we simplify both sides of the equation:

5 * (12y - 1) = 2 * (9y + 8)

3. Distribute the Terms

Next, we distribute the numbers on both sides of the equation:

60y - 5 = 18y + 16

4. Combine Like Terms

Now, we want to get all the y terms on one side of the equation and the constants on the other side. We can subtract 18y from both sides:

60y - 18y - 5 = 18y - 18y + 16

42y - 5 = 16

Next, we add 5 to both sides:

42y - 5 + 5 = 16 + 5

42y = 21

5. Isolate y

Finally, we isolate y by dividing both sides by 42:

y = 21 / 42

y = 1/2

So, the value of y that makes the equation true is 1/2.

Detailed Explanation of Each Step

To ensure a thorough understanding, let's revisit each step with a more detailed explanation:

Eliminating Fractions: Multiplying by the LCM

The presence of fractions can make an equation look daunting, but eliminating them simplifies the process significantly. We identified the denominators as 2 and 5. The least common multiple (LCM) is the smallest number that both 2 and 5 divide into evenly. In this case, the LCM is 10. By multiplying both sides of the equation by 10, we ensure that the denominators cancel out, leaving us with a simpler equation to solve. This step is crucial because it transforms the equation from a fractional form to a more manageable linear form.

Simplifying the Equation: Distributive Property

After multiplying by the LCM, we obtain 5 * (12y - 1) = 2 * (9y + 8). The next step involves applying the distributive property, which states that a(b + c) = ab + ac. This means we multiply the number outside the parentheses by each term inside the parentheses. On the left side, 5 is multiplied by both 12y and -1, resulting in 60y - 5. On the right side, 2 is multiplied by both 9y and 8, resulting in 18y + 16. This step expands the equation and prepares it for the next phase of isolating the variable y.

Combining Like Terms: Isolating the Variable

Combining like terms is a critical step in solving algebraic equations. The goal is to group the terms containing the variable y on one side of the equation and the constant terms on the other side. We began by subtracting 18y from both sides of the equation (60y - 5 = 18y + 16) to move the y term from the right side to the left side. This resulted in 42y - 5 = 16. Next, we added 5 to both sides to move the constant term from the left side to the right side, which gave us 42y = 21. By performing these operations, we have successfully isolated the y term on one side of the equation.

Isolating y: Final Division

The final step in solving for y is to divide both sides of the equation by the coefficient of y. In our equation, 42y = 21, the coefficient of y is 42. Dividing both sides by 42 isolates y and gives us the solution. So, y = 21 / 42, which simplifies to y = 1/2. This concludes the process of solving for y, and we have found the value that satisfies the original equation.

Alternative Methods to Solve the Equation

While the step-by-step method outlined above is a standard approach, there are alternative methods to solve the equation (12y - 1)/2 = (9y + 8)/5. One such method involves cross-multiplication, which can be a quicker way to eliminate fractions in an equation. Let's explore this method.

Cross-Multiplication Method

Cross-multiplication is a technique used to solve equations involving fractions. It is based on the principle that if a/b = c/d, then ad = bc. In our equation (12y - 1)/2 = (9y + 8)/5, we can apply cross-multiplication by multiplying the numerator of the left side by the denominator of the right side, and vice versa. This will eliminate the fractions in one step.

Applying Cross-Multiplication

To apply cross-multiplication to the equation (12y - 1)/2 = (9y + 8)/5, we multiply (12y - 1) by 5 and (9y + 8) by 2. This gives us:

5 * (12y - 1) = 2 * (9y + 8)

Notice that this is the same equation we obtained after multiplying both sides by the LCM in the step-by-step method. From here, we can continue with the steps of distributing the terms, combining like terms, and isolating y, as we did before.

Comparison with the LCM Method

Cross-multiplication is essentially a shortcut that combines the first two steps of the LCM method. It directly eliminates the fractions without explicitly finding the LCM. However, it is important to note that cross-multiplication is only applicable when you have a proportion, i.e., a single fraction equal to another single fraction. If there are multiple fractions or terms on either side of the equation, the LCM method is generally more versatile.

Advantages and Disadvantages

Advantages of Cross-Multiplication:

  • Efficiency: It can be quicker for simple equations with only two fractions.
  • Direct: It eliminates the need to find the LCM explicitly.

Disadvantages of Cross-Multiplication:

  • Limited Applicability: It is only suitable for equations in the form a/b = c/d.
  • Potential for Errors: Without understanding the underlying principles, it can be applied incorrectly.

When to Use Cross-Multiplication

Cross-multiplication is most useful when dealing with equations where a single fraction is set equal to another single fraction. It simplifies the initial steps and can save time. However, for more complex equations with multiple fractions or terms, the LCM method provides a more systematic and reliable approach.

Common Mistakes to Avoid

Solving algebraic equations can be tricky, and it's easy to make mistakes if you're not careful. Here are some common mistakes to watch out for when solving equations like (12y - 1)/2 = (9y + 8)/5:

1. Incorrectly Distributing Terms

A common mistake is to incorrectly apply the distributive property. Remember that the number outside the parentheses must be multiplied by every term inside the parentheses. For example, in the equation 5 * (12y - 1) = 2 * (9y + 8), it's essential to multiply 5 by both 12y and -1, and 2 by both 9y and 8. A mistake here can lead to an incorrect equation and a wrong solution.

2. Not Multiplying All Terms by the LCM

When eliminating fractions, it's crucial to multiply every term on both sides of the equation by the LCM. For instance, if you forget to multiply a constant term, the equation will be unbalanced, and the solution will be incorrect. This oversight can often occur when dealing with more complex equations with multiple terms.

3. Combining Unlike Terms Incorrectly

Combining like terms is a fundamental step, but it's essential to combine only terms that are alike. For example, you can combine 60y and -18y because they both have the variable y, but you cannot combine 60y with -5 because -5 is a constant term. Mixing up like and unlike terms will lead to an incorrect simplification of the equation.

4. Incorrectly Applying Operations to Both Sides

The golden rule of solving equations is that whatever operation you perform on one side, you must perform on the other side to maintain equality. If you add a number to one side but forget to add it to the other side, the equation becomes unbalanced. Similarly, if you divide only one side by a number, the equality is lost. Always ensure that any operation is applied consistently to both sides.

5. Forgetting to Distribute the Negative Sign

When dealing with negative signs, it's easy to make a mistake. For example, if you have an expression like -(3y - 2), you need to distribute the negative sign to both terms inside the parentheses, making it -3y + 2. Forgetting to distribute the negative sign correctly can lead to an incorrect equation and a wrong solution.

6. Not Simplifying Fractions Correctly

After solving for y, you might end up with a fraction that needs simplification. For instance, if you get y = 21/42, you need to simplify this fraction to y = 1/2. Failing to simplify fractions can lead to a correct answer that is not in its simplest form.

7. Making Arithmetic Errors

Simple arithmetic errors can derail the entire solution process. Whether it's a mistake in addition, subtraction, multiplication, or division, an arithmetic error can lead to an incorrect answer. Double-checking your calculations can help catch these errors before they cause problems.

8. Not Checking the Solution

Finally, one of the best ways to avoid mistakes is to check your solution. Plug the value you found for y back into the original equation to see if it holds true. If both sides of the equation are equal, then your solution is correct. If not, you know there's a mistake somewhere, and you need to go back and review your steps.

Real-World Applications of Solving Linear Equations

Solving linear equations like (12y - 1)/2 = (9y + 8)/5 is not just an abstract mathematical exercise; it has numerous real-world applications. Understanding how to manipulate and solve these equations is crucial in various fields, from basic budgeting to complex engineering problems. Let's explore some practical applications.

1. Budgeting and Personal Finance

Linear equations are fundamental in personal finance. For example, suppose you want to save a certain amount of money each month. You can set up a linear equation to determine how much you need to save per month to reach your goal within a specific timeframe. Similarly, understanding loan calculations, such as determining monthly payments for a car or mortgage, often involves solving linear equations.

2. Business and Economics

In business, linear equations are used for cost analysis, revenue projection, and break-even analysis. For instance, a company might use a linear equation to model the relationship between the number of units sold and the total revenue. By solving this equation, they can determine the number of units they need to sell to cover their costs and start making a profit. Economic models often use systems of linear equations to analyze market trends and predict economic behavior.

3. Engineering and Physics

Engineering and physics rely heavily on mathematical models, many of which involve linear equations. Engineers use these equations to design structures, analyze circuits, and model fluid dynamics. Physicists use linear equations to describe motion, energy, and forces. For example, calculating the trajectory of a projectile or determining the current in an electrical circuit involves solving linear equations.

4. Chemistry

In chemistry, linear equations are used for balancing chemical equations and calculating reaction rates. Balancing chemical equations ensures that the number of atoms of each element is the same on both sides of the equation, which is a fundamental principle of chemistry. Reaction rates can be modeled using linear equations to predict how quickly a chemical reaction will proceed under different conditions.

5. Computer Science

Linear equations are also used in computer science, particularly in areas like computer graphics and machine learning. In computer graphics, linear transformations are used to manipulate images and create animations. In machine learning, linear regression models are used to predict outcomes based on input data. Understanding linear equations is essential for developing algorithms and solving computational problems.

6. Everyday Problem Solving

Beyond these specialized fields, linear equations can help solve everyday problems. For example, if you're planning a road trip, you can use a linear equation to calculate how much gasoline you'll need based on the distance you'll travel and the fuel efficiency of your car. Similarly, if you're cooking and need to adjust a recipe for a different number of servings, you might use linear equations to scale the ingredients.

Conclusion

In conclusion, finding the value of y in the equation (12y - 1)/2 = (9y + 8)/5 is a fundamental algebraic problem that highlights the importance of understanding how to manipulate equations. By following a step-by-step approach, we were able to isolate y and determine that y = 1/2 is the solution. We also explored alternative methods, such as cross-multiplication, and discussed common mistakes to avoid. Furthermore, we examined the real-world applications of solving linear equations, emphasizing their relevance in various fields. Mastering these skills is essential for anyone pursuing further studies in mathematics, science, or engineering, as well as for everyday problem-solving. The ability to solve linear equations empowers individuals to analyze situations, make informed decisions, and tackle challenges effectively.