Problem 36 - Algebra 1: Concepts and Skills

Question

Solve the equation. \((y-5)(y-6)(3 y-2)=0\)

Step-by-Step Solution

Answer

The solutions to the equation are \(y = 5\), \(y = 6\), and \(y = \frac{2}{3}\).

Step 1: Apply the Zero-Product Property

Set each factor equal to zero and solve for \(y\). Thus, we have following three equations:\(y - 5 = 0\)\(y - 6 = 0\)\(3y - 2 = 0\)

Step 2: Solve Each Equation

Add 5, 6 and \(\frac{2}{3}\) respectively to each side of each equation to solve for \(y\).With \(y - 5 = 0\), add 5 to both sides, thus \(y = 5\).For \(y - 6 = 0\), add 6 to both sides, thus \(y = 6\).Lastly, for \(3y - 2 = 0\), add 2 to both sides and then divide by 3, thus \(y = \frac{2}{3}\).

Step 3: State the Solution

The solutions to the equation are \(y = 5\), \(y = 6\), and \(y = \frac{2}{3}\).

Key Concepts

Solving Linear Equations

Solving linear equations is like finding the right key for a lock. When you have an equation, it represents a balance. Your job is to keep both sides of the equation equal while you figure out the value of the unknown variable, often shown as "y" or "x". In the original exercise, we had a product of several factors set to zero. This scenario is a common one where we utilize the zero-product property to find the solution.

In most linear equations, you’ll follow a series of straightforward steps:

• Identify the equation you need to solve.
• Isolate the variable by performing addition, subtraction, multiplication, or division on both sides. This maintains the balance.
• Solve for the variable. This may happen in one or several steps.

Your aim is to solve for the variable in the simplest way so you can clearly see what value makes the equation true. Once you have found the solution, you can double-check your work by substituting the found value back into the original equation to see if it indeed balances.

Factoring

Factoring is the process of breaking down an expression into products of simpler factors. Think of it as unmasking a complex component into its constituent parts. In the context of the given problem, factoring allows us to solve equations by setting each factor equal to zero. This is thanks to the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero.

When you factor expressions, remember these tips:

• Look for common factors in algebraic terms to simplify the equation.
• Consider special patterns, such as difference of squares or trinomials, for easier factoring.
• Check using multiplication to ensure your factors multiply back to the original product.

By factoring correctly, you're able to split a complicated expression into manageable parts and solve for the variable bit by bit.

Algebraic Equations

Algebraic equations are mathematical statements involving variables, constants, and operational symbols. They are the building blocks of algebra and essential tools for solving a variety of mathematical problems. In algebraic equations like the one in the original exercise, you’re dealing with one or multiple unknowns linked by mathematical operations.

To get a better grasp on algebraic equations, it's helpful to:

• Understand the terms: Coefficients (numbers in front of variables), constants (stand-alone numbers), and variables themselves.
• Practice translating word problems into algebraic expressions and equations.
• Learn properties of equality and inverse operations to manipulate and solve equations.

Algebraic equations are a bit like puzzles. The aim is to find the value of variables that satisfy the entire equation. Engaging with these problems helps sharpen mathematical thinking and problem-solving skills.