Problem 372 - Intermediate Algebra

Question

In the following exercises, simplify using the Distributive Property. $$ (4 y-1)-(y-2) $$

Step-by-Step Solution

Answer

3y + 1

Step 1 - Distribute the subtraction

Apply the subtraction to each term inside the parentheses. Rewrite the expression as (4y - 1) - (y) - (-2)

Step 2 - Simplify the signs in the expression

Rewrite -(-2) as +2 (because subtracting a negative is the same as adding a positive). The expression now becomes 4y - 1 - y + 2

Step 3 - Combine like terms

Combine the terms with y: 4y - y = 3y and the constant terms: -1 + 2 = 1 The final expression is 3y + 1

Key Concepts

Combining Like Terms

To simplify an algebraic expression, one important step is to combine like terms. Like terms are terms that have the same variables raised to the same powers. For example, in the expression \(4y - y\), both terms are like terms because they both contain the variable \(y\).
When combining like terms:

• Add or subtract the coefficients of the terms. Keeping their variables unchanged.

In our exercise, we combined \(4y\) and \(y\). This gave us \(4y - y = 3y\).
Combining like terms is crucial as it simplifies the expression and makes it more manageable.

Distributive Property

The Distributive Property allows you to multiply a sum or difference by distributing the multiplication to each term within the parentheses. This property is expressed as: \(a(b + c) = ab + ac\)
In the given exercise, we see \((4y-1)-(y-2)\).
We distribute the subtraction to both terms inside the parentheses:

• We rewrite it as \((4y-1) - y - (-2)\).

Notice that \(-(-2)\) becomes \(+2\), giving us \(4y-1-y+2\). The Distributive Property helps us simplify the expression by removing parentheses and clarifying which terms must be combined.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as simple as possible. This often involves:

• Applying the Distributive Property
• Combining like terms
• Eliminating any unnecessary parentheses

For the exercise \((4y-1)-(y-2)\), we distributed the subtraction and combined like terms, turning it into \(3y + 1\).
Each step reduces the complexity of the expression and brings us closer to the most simplified form. Simplifying expressions is essential for solving equations and understanding algebraic expressions better.

Negative Numbers

Negative numbers represent values less than zero. When simplifying expressions, it's important to handle negative signs correctly.
In our exercise, we dealt with \(-(y-2)\):

• Opening the parentheses applied the subtraction to each term inside: \(-y - (-2)\)
• Then, we remembered that \(-(-2)\) is the same as \(+2\).

This step of changing \(-(-2)\) to \(+2\) highlights the importance of precision when working with negative numbers. Handling negative numbers correctly ensures that our simplified expression is accurate and error-free.