Problem 66 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 4 y[5-(y+1)]+3 y(y+1) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(15y - y^2\).

Step 1: Apply Distributive Property

First, we apply the distributive property to remove the symbols of grouping. Distributive law states that for all real numbers a, b, and c: \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\) . Therefore, the given expression changes like this: 4y[5 - (y + 1)] + 3y(y + 1) will become \(= 4y*5 - 4y*(y + 1) + 3y*y + 3y*1\) \(= 20y - 4y^2 - 4y + 3y^2 + 3y\)

Step 2: Combine Like Terms

Now, we'll combine the like terms. This involves adding or subtracting coefficients of the terms with same variables. So, we can further simplify the expression to: \(= 20y - 4y - 4y + 3y + 3y^2 - 4y^2\) \(= (20 - 4 - 4 + 3)y + (3 - 4)y^2\) \(= 15y - y^2\)

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra that helps simplify expressions and solve equations. It basically states that when you distribute a multiplication over an addition or subtraction within brackets, you multiply each term inside the bracket by the term outside. For example, when you see something like \(a(b + c)\), you can expand it as \(ab + ac\). This is because you're distributing the \(a\) to both \(b\) and \(c\).
In the provided exercise, you apply this property twice:

• First, to handle the term \(4y[5 - (y + 1)]\). You distribute the \(4y\) into the bracket, which turns the expression into \(4y*5 - 4y*(y + 1)\).
• Then, for the term \(3y(y + 1)\), you distribute \(3y\) to both \(y\) and \(+1\), turning it into \(3y*y + 3y*1\).

The result of successfully using the distributive property here is that the expression is stripped of its grouping symbols, which is an essential first step toward simplification.

Combining Like Terms

Once you have used the distributive property, the next step is to combine like terms. Like terms are terms that have the same variables and the same powers. This means you can add or subtract their coefficients while keeping the variable part unchanged.
In the exercise, the expanded expression becomes \(20y - 4y^2 - 4y + 3y^2 + 3y\). Look for terms with the same variable structure:

• Like terms for \(y\) are \(20y, -4y, \) and \(3y\).
• Like terms for \(y^2\) are \(3y^2\) and \(-4y^2\).

Adding the coefficients gives you:

• \(20y - 4y + 3y = 19y\).
• \(3y^2 - 4y^2 = -y^2\).

So, after combining these like terms, the expression simplifies down to \(19y - y^2\). This process is crucial for neatly expressing and working with algebraic equations.

Algebraic Manipulation

Algebraic manipulation refers to the variety of techniques used to transform and solve algebraic expressions and equations. The ultimate aim is to simplify expressions or solve equations for a particular variable. It encompasses methods such as multiplying, factoring, expanding, and simplifying terms in an expression.
In the exercise, you performed a few key manipulations:

• Using the distributive property to expand terms that were initially inside brackets.
• Rearranging terms to line up like terms next to each other in preparation for combining them.
• Correctly identifying and simplifying the expression by subtracting or adding similar quantities.

These manipulations are instrumental in ensuring that complex expressions are manageable and straightforward, allowing us to further analyze or input these expressions into larger systems like equations.

Grouping Symbols in Algebra

Grouping symbols in algebra, such as parentheses \(()\), brackets \([]\), and braces \({}\), are critical for determining the order of operations in complex algebraic expressions. They tell us which operations should be performed first and help clarify how parts of expressions are related.
In the given exercise, the brackets around \([5 - (y + 1)]\) served as our first signal that we needed to simplify what's inside before moving further. The use of grouping symbols indicates a priority in operations by suggesting an enclosed set of terms that should be treated as a single unit.

• Always start simplifying from the innermost grouping symbol.
• Apply operations (like the distributive property) to remove these symbols.
• Once removed, proceed with usual algebraic processes like combining like terms.

Understanding grouping symbols allows for precise and accurate evaluation of expressions, ensuring expressions are manipulated in the correct order and leading to the correct solution.