Problem 66 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ (y+47)(y-27)=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(y = -47\) and \(y = 27\).

Step 1: Understand the Zero Product Property

In any equation, if the product of two factors equals zero, then at least one of the factors must be zero. In other words, if (y + 47) * (y - 27) = 0, either (y + 47) = 0 or (y - 27) = 0.

Step 2: Solve for y in the first factor

Let's start by setting (y + 47) = 0. Solving for \(y\), you subtract 47 from both sides to get \(y = -47\). That's your first solution.

Step 3: Solve for y in the second factor

Next, set (y - 27) equal to zero. Adding 27 to both sides, you get \(y = 27\). That's your second solution.

Key Concepts

Zero Product Property

Equation solving often involves breaking down complex expressions into manageable parts, and one key strategy is using the Zero Product Property. This principle tells us that if the product of two terms is zero, then one or both terms must be zero. For instance, if you have an equation like \( (a)(b) = 0 \), it tells us that either \( a = 0 \) or \( b = 0 \). This can simplify solving problems significantly.

• This property applies to situations where two or more factors multiply to zero.
• It allows us to split the equation into separate, simpler equations.

Understanding the Zero Product Property is crucial as it forms the basis for solving many polynomial equations where the expression can be written as a product of factors. Once you identify the factors, each can be set to zero, one at a time, to find the possible solutions.

Factoring Equations

To apply the Zero Product Property effectively, you must first factor the equation. Factoring is the process of breaking down an equation into simpler components that are multiplied together. In algebra, this often means converting a polynomial into a product of simpler polynomials. For example, \( (y+47)(y-27)=0 \) is already factored, which makes it simple to apply the Zero Product Property.

• Factoring can involve identifying common factors among terms.
• You'll often see the use of techniques like grouping or using formulas for special products (e.g., difference of squares).

The goal of factoring is to rewrite the equation so that the Zero Product Property can be applied to find solutions. Once an equation is factored, each part can be tackled separately, as each factor can independently equal zero, leading to potential solutions.

Algebraic Solutions

Finding algebraic solutions involves isolating the variable to find its value(s). With a factored equation like \( (y+47)(y-27)=0 \), the process becomes straightforward. For each factor, set the expression to zero and solve for the variable.

• Start with each factor: \( y+47=0 \) and \( y-27=0 \).
• For \( y+47=0 \), subtract 47 from both sides to find \( y=-47 \).
• For \( y-27=0 \), add 27 to both sides to find \( y=27 \).

This simple step-by-step approach ensures you find all possible solutions. Each provides a different value for the variable satisfying the original equation. Practicing these steps strengthens your ability to solve more complex algebraic equations confidently.