Solving (3y - 5)(8 + Y) = 0 A Step-by-Step Guide

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In mathematics, solving algebraic equations is a fundamental skill. Algebraic equations are mathematical statements that assert the equality of two expressions. Solving an equation means finding the value(s) of the variable(s) that make the equation true. There are various techniques for solving different types of equations, and the best approach depends on the specific structure of the equation. One common type of equation is a polynomial equation, which involves terms with variables raised to non-negative integer powers. The degree of a polynomial equation is the highest power of the variable in the equation. For instance, a linear equation has a degree of 1, a quadratic equation has a degree of 2, and so on. The equation we are going to solve in this article, (3y - 5)(8 + y) = 0, is a quadratic equation, although it's presented in a factored form. This form offers a direct way to find the solutions, leveraging the Zero Product Property. This property is a cornerstone in solving equations where a product of factors equals zero. It states that if the product of two or more factors is zero, then at least one of the factors must be zero. This simple yet powerful principle allows us to break down the problem into simpler parts, solving each factor separately to find the overall solutions of the equation. The ability to solve equations is crucial in many areas of mathematics and its applications, including physics, engineering, economics, and computer science. Therefore, understanding the methods for solving different types of equations is an essential skill for students and professionals alike. In the following sections, we will delve deeper into how to apply the Zero Product Property to solve the given equation and explore the broader implications of this method in problem-solving.

Understanding the Zero Product Property

The Zero Product Property is a fundamental concept in algebra that provides a straightforward method for solving equations where a product of factors equals zero. This property states that if the product of two or more factors is zero, then at least one of the factors must be equal to zero. Mathematically, this can be expressed as follows: If A * B = 0, then either A = 0 or B = 0 (or both). This principle is particularly useful when dealing with equations that are factored or can be factored, as it allows us to break down a complex equation into simpler equations that are easier to solve. For example, consider the equation (x - 2)(x + 3) = 0. According to the Zero Product Property, this equation is true if either (x - 2) = 0 or (x + 3) = 0. Solving these two simpler equations gives us x = 2 and x = -3, which are the solutions to the original equation. The Zero Product Property is not only a powerful tool for solving equations but also a cornerstone in understanding the behavior of polynomial functions. It helps us find the roots or zeros of a polynomial, which are the values of the variable that make the polynomial equal to zero. These roots correspond to the x-intercepts of the graph of the polynomial function, providing valuable information about the function's behavior. In addition to its application in solving equations, the Zero Product Property is also used in various mathematical proofs and theoretical arguments. It serves as a building block for more advanced concepts and techniques in algebra and calculus. Therefore, a solid understanding of this property is essential for anyone studying mathematics or related fields. The equation we aim to solve, (3y - 5)(8 + y) = 0, perfectly illustrates the application of the Zero Product Property. By recognizing that the equation is a product of two factors equaling zero, we can easily find the solutions by setting each factor equal to zero and solving for the variable. This approach not only simplifies the solution process but also highlights the elegance and efficiency of the Zero Product Property in solving algebraic equations.

Step-by-Step Solution of (3y - 5)(8 + y) = 0

To solve the equation (3y - 5)(8 + y) = 0, we will apply the Zero Product Property. This property allows us to break down the equation into simpler parts, making it easier to find the solutions. The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In this case, we have two factors: (3y - 5) and (8 + y). According to the property, the equation (3y - 5)(8 + y) = 0 is true if either (3y - 5) = 0 or (8 + y) = 0 (or both). Let's solve each of these equations separately:

  1. Solve the first factor:

    • 3y - 5 = 0
    • To isolate the variable y, we first add 5 to both sides of the equation:
    • 3y - 5 + 5 = 0 + 5
    • 3y = 5
    • Next, we divide both sides by 3 to solve for y:
    • 3y / 3 = 5 / 3
    • y = 5/3

  2. Solve the second factor:

    • 8 + y = 0
    • To isolate y, we subtract 8 from both sides of the equation:
    • 8 + y - 8 = 0 - 8
    • y = -8

Therefore, the solutions to the equation (3y - 5)(8 + y) = 0 are y = 5/3 and y = -8. We can verify these solutions by substituting them back into the original equation to ensure that they make the equation true. This step-by-step solution demonstrates how the Zero Product Property simplifies the process of solving factored equations. By breaking down the equation into smaller, more manageable parts, we can easily find the values of the variable that satisfy the equation. This method is a fundamental tool in algebra and is widely used in solving various types of equations.

Verifying the Solutions

After solving an equation, it is always a good practice to verify the solutions. This step helps to ensure that the solutions are correct and that no errors were made during the solving process. To verify the solutions, we substitute each solution back into the original equation and check if the equation holds true. For the equation (3y - 5)(8 + y) = 0, we found two solutions: y = 5/3 and y = -8. Let's verify each solution:

  1. Verify y = 5/3:

    • Substitute y = 5/3 into the original equation:
    • (3(5/3) - 5)(8 + 5/3) = 0
    • Simplify the first factor:
    • (5 - 5)(8 + 5/3) = 0
    • 0 * (8 + 5/3) = 0
    • Since the first factor is 0, the entire expression is 0, which satisfies the equation.

  2. Verify y = -8:

    • Substitute y = -8 into the original equation:
    • (3(-8) - 5)(8 + (-8)) = 0
    • Simplify the second factor:
    • (3(-8) - 5)(0) = 0
    • Since the second factor is 0, the entire expression is 0, which satisfies the equation.

Both solutions, y = 5/3 and y = -8, satisfy the original equation. This verification process confirms that our solutions are correct. Verifying solutions is a crucial step in problem-solving, especially in mathematics. It not only helps to catch any errors but also reinforces the understanding of the concepts and techniques used in solving the equation. By substituting the solutions back into the original equation, we can see how the equation works and why the solutions are valid. This practice can also help to build confidence in one's problem-solving abilities. In addition to verifying solutions algebraically, it is also possible to verify them graphically. For example, if we were solving for the roots of a function, we could graph the function and see where it intersects the x-axis. The x-intercepts would correspond to the solutions of the equation. This graphical verification can provide a visual confirmation of the solutions and further enhance understanding.

Implications and Applications

Solving the equation (3y - 5)(8 + y) = 0 using the Zero Product Property is not just a mathematical exercise; it has significant implications and applications in various fields. Understanding how to solve such equations is crucial for students studying algebra and higher-level mathematics. The Zero Product Property is a fundamental concept that underlies many algebraic techniques, including factoring polynomials, solving quadratic equations, and finding the roots of polynomial functions. In calculus, for example, finding the zeros of a function is often a critical step in analyzing its behavior, such as determining where the function changes direction or where it has maximum or minimum values. The ability to solve equations is also essential in many real-world applications. In physics, equations like this might arise when modeling the trajectory of a projectile or analyzing the equilibrium of forces. In engineering, they could be used to design structures or analyze electrical circuits. In economics, they might appear in models of supply and demand or in financial calculations. For instance, consider a simple model of supply and demand where the quantity demanded (Qd) and the quantity supplied (Qs) are given by the equations Qd = 100 - 2P and Qs = 3P - 50, where P is the price. To find the equilibrium price, we set Qd equal to Qs and solve for P. This often involves solving an equation similar in form to the one we solved earlier. Furthermore, the skills developed in solving algebraic equations are transferable to other areas of problem-solving. The ability to break down a complex problem into smaller, more manageable parts, to identify the key relationships and constraints, and to apply logical reasoning are all valuable skills that can be used in a wide range of contexts. In summary, the seemingly simple task of solving an equation like (3y - 5)(8 + y) = 0 has far-reaching implications and applications. It is a fundamental skill that is essential for success in mathematics and many other fields. By mastering this skill, students can develop a deeper understanding of mathematical concepts and improve their problem-solving abilities.

In conclusion, we have successfully solved the equation (3y - 5)(8 + y) = 0 by applying the Zero Product Property. This property allowed us to break down the equation into two simpler equations, 3y - 5 = 0 and 8 + y = 0, which we then solved individually to find the solutions y = 5/3 and y = -8. We also verified these solutions by substituting them back into the original equation, confirming their correctness. The Zero Product Property is a powerful tool for solving equations where a product of factors equals zero. It is a fundamental concept in algebra and is used extensively in solving various types of equations, including polynomial equations, quadratic equations, and more complex algebraic expressions. The ability to solve equations is a crucial skill in mathematics and has numerous applications in other fields, such as physics, engineering, economics, and computer science. By mastering the techniques for solving equations, students can develop a deeper understanding of mathematical concepts and improve their problem-solving abilities. The step-by-step approach we used in this article demonstrates a systematic way to solve equations. First, we identified the structure of the equation and recognized that it was in a factored form. Then, we applied the Zero Product Property to break down the equation into simpler parts. Next, we solved each of the simpler equations separately. Finally, we verified the solutions to ensure their correctness. This systematic approach can be applied to solving a wide range of equations and problems. In addition to solving equations, it is also important to understand the underlying concepts and principles. The Zero Product Property, for example, is based on the fundamental properties of numbers and operations. By understanding these properties, we can better understand why the Zero Product Property works and how to apply it effectively. Overall, solving the equation (3y - 5)(8 + y) = 0 is a valuable exercise that reinforces key algebraic concepts and techniques. It demonstrates the power of the Zero Product Property and highlights the importance of systematic problem-solving. By mastering these skills, students can build a strong foundation for further study in mathematics and related fields.