Problem 24 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ y(3 y-4)=0 $$

Step-by-Step Solution

Answer

Answer: The values of y that satisfy the given equation are y = 0 and y = \frac{4}{3}.

Step 1: Identify the factors in the equation

The given equation has two factors: y and (3y - 4).

Step 2: Apply Zero-product Property

According to the zero-product property, either y = 0, or (3y - 4) = 0, or both. We will now solve for y in each case.

Step 3: Solve y = 0

In this case, the solution is already provided: y = 0.

Step 4: Solve (3y - 4) = 0 for y

To solve for y, we will first isolate y by adding 4 to both sides of the equation: 3y - 4 + 4 = 0 + 4 This simplifies to: 3y = 4 Now, divide both sides by 3: y = \frac{4}{3}

Step 5: State the solution set

The solution set of the given equation contains the values of y that satisfy the zero-product property, which are y = 0 and y = \frac{4}{3}.

Key Concepts

Zero-product Property

The zero-product property is a fundamental principle in algebra that helps us solve equations involving products of terms. This property states that if the product of two or more factors equals zero, then at least one of these factors must be zero.
This concept can be visualized easily:

• If \(a \times b = 0\), then either \(a = 0\), \(b = 0\), or both.

In our exercise, the equation \(y(3y-4) = 0\) exemplifies this property. To find the solutions, we determine which factors (either \(y\) or \(3y-4\)) could be zero. This method simplifies the process of finding exact solutions for polynomial equations by breaking them into smaller parts.

Equation Solving

Equation solving involves finding the values of variables that make an algebraic equation true. When solving equations like \(y(3y-4) = 0\), we utilize properties like zero-product property to simplify the process.
The basic steps for solving equations include:

• Identifying the factors involved in the equation.
• Applying the appropriate mathematical property, such as the zero-product property.
• Solving each factor by isolating the variable.

By approaching the equation step-by-step, we can systematically find the solutions that satisfy the equation. This process teaches us not just how to solve a particular equation, but the foundational skills needed to tackle more complex equations.

Factorization

Factorization is the process of breaking down an expression into a product of its simpler factors. This method is especially useful when dealing with polynomial equations, where rewriting the equation as a product of factors can significantly simplify solving it.
In our exercise, the equation \(y(3y-4) = 0\) is already presented in its factored form.
To factor an equation means to express it as a product of terms, such as:

• From something like \(x^2 - 4x\) to \(x(x-4)\).
• To confirm factorization, you can expand the factors back out to ensure they reconstruct the original equation.

Successfully factorizing an equation allows us to apply the zero-product property effectively, making it easier to identify possible solutions.

Solution Set

The solution set of an equation is the collection of all possible values of the variable that satisfy the equation. In algebraic terms, these solutions are the numbers that make the mathematical statement true when substituted back into the original equation.
For the given equation \(y(3y-4) = 0\), we determine the solution set by solving the factors:

• Setting \(y = 0\) directly results in one part of the solution.
• Solving \(3y-4=0\) yields \(y=\frac{4}{3}\) as another solution.

Hence, the complete solution set for this equation is \( \{0, \frac{4}{3}\} \). Understanding and determining the solution set is crucial because it clearly articulates all the answers derived from solving the equation and represents the full set of solutions applicable to the problem context.