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Chapter 4: Problem 117

Simplify or solve as appropriate. $$3 y(y+2)=3(y+1)(y-1)$$

Short Answer

Expert verified
The solution to the equation is \(y = -0.5\).

Step by step solution

01

Expand both sides of the equation

Apply distributive property on both sides of the equation to expand them. This gives: \(3y^2 + 6y = 3y^2 - 3\).
02

Simplify the equation

Next, subtract \(3y^2\) from both sides of the equation to simplify it. This leaves: \(6y = -3\).
03

Solve for y

Finally, to solve for 'y', divide both sides of the equation by 6. This gives: \(y = -3/6 = -0.5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
When working with algebraic equations, the distributive property is a fundamental principle that helps simplify expressions and solve equations. The distributive property involves multiplying a single term by each term inside a set of parentheses. This is especially useful in simplifying complex expressions. For instance, if you have an expression like \(3y(y+2)\), you can use the distributive property to expand it to \(3y \times y + 3y \times 2\), which results in \(3y^2 + 6y\).
  • The property applies to both addition and subtraction: \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\).
  • Using the distributive property can help break down complex problems into simpler parts, making them easier to handle.
By applying this property on both sides of the equation in our exercise, we expanded everything into simpler parts. This was the critical first step in approaching the solution.
Solving Equations
Solving equations means finding the value of the unknown variable that makes the equation true. Once you've got your expression expanded using the distributive property, the next phase focuses on solving for the variable at hand.
The goal is often to isolate the variable on one side of the equation. In our specific example, after applying the distributive property, we got the equation \(3y^2 + 6y = 3y^2 - 3\). The next step was to simplify this by isolating the variable term \(6y\) after applying appropriate operations.
  • The first action taken was subtracting \(3y^2\) from both sides, aiming to eliminate \(3y^2\), which simplifies the equation to \(6y = -3\).
  • To isolate \(y\), you divide both sides by 6, getting \(y = -0.5\).
This sequential approach of rearranging and simplifying is key to solving equations effectively, as it ensures the solution satisfies the original equation.
Simplifying Expressions
Simplifying expressions involves making them easier to work with while retaining their original value. This step is crucial in algebra as it prepares an equation for solving. Simplifying involves reducing terms, combining like terms, and performing arithmetic operations.
In our case, after using the distributive property on \(3y(y+2)\) and \(3(y+1)(y-1)\), we expanded to \(3y^2 + 6y\) and \(3y^2 - 3\) respectively.
  • The simplification step helped in reducing the complexity by canceling out \(3y^2\) from both sides, leading to a more manageable equation \(6y = -3\).
  • Once simplified, the expression was much easier to solve for \(y\).
Getting to a point where the expressions are straightforward makes problem-solving faster and minimizes errors, essential for achieving accurate results.

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