Problem 316 - Elementary Algebra

Question

Solve. \((y-11)(y+1)=0\)

Step-by-Step Solution

Answer

y = 11 or y = -1

Step 1: Understand the Zero Product Property

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This means either ewline (y-11) = 0 or (y+1) = 0.

Step 2: Solve the First Equation

Set the first factor equal to zero and solve for y: ewline (y-11) = 0 ewline y = 11

Step 3: Solve the Second Equation

Set the second factor equal to zero and solve for y: ewline (y+1) = 0 ewline y = -1

Step 4: State the Solutions

The solutions to the equation ewline (y-11)(y+1) = 0 ewline are y = 11 and y = -1.

Key Concepts

Factoring Equations

Factoring equations is essential for solving various types of algebraic problems. It involves rewriting an equation as a product of its factors. For instance, in the given exercise, the equation \( (y-11)(y+1)=0 \) is already factored.
Each factor represents a simpler equation that can be solved individually. Understanding how to factor an equation allows you to break down more complex problems into manageable parts. Common methods for factoring include:

• Identifying common factors
• Using the distributive property
• Applying specific formulas (e.g., difference of squares)

Solving Quadratic Equations

Quadratic equations often take the form \( ax^2 + bx + c = 0 \). These can be solved using methods such as factoring, completing the square, or utilizing the quadratic formula. In our exercise, we solve a quadratic equation by factoring, which reveals the solutions directly.

• First, we factor the equation: \((y-11)(y+1)=0\)
• Next, we apply the zero product property to find \(y-11=0\) or \(y+1=0\)
• Finally, we solve these simpler equations: \(y=11\) and \(y=-1\)

This method is efficient when the quadratic is easily factorable.

Algebraic Properties

Understanding algebraic properties is crucial for solving equations and manipulating expressions. Here are a few key properties that are often used:

Distributive Property: This property states that \(a(b + c) = ab + ac\). It is useful for expanding and factoring expressions.

Commutative Property: This property means that the order of addition or multiplication does not affect the result, i.e., \(a + b = b + a\) and \(ab = ba\).

Associative Property: This property suggests grouping does not impact the result, such as \((a + b) + c = a + (b + c)\).

These fundamental properties make it easier to understand more complex operations and solve equations like the one in our example.