Problem 48 - College Algebra

Question

For the following exercises, simplify the expression. $$ 18 y-2(1+7 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(4y - 2\).

Step 1: Distribute the -2

Start by distributing the -2 to each term inside the parentheses. This involves multiplying -2 by each term within the parentheses. The expression becomes:\[18y - 2(1) - 2(7y)\] which simplifies to \[18y - 2 - 14y\].

Step 2: Combine like terms

Now, combine the like terms in the expression. Look at the terms involving \(y\), which are \(18y\) and \(-14y\). Combine these by subtraction: \[18y - 14y = 4y\]. The expression now becomes \[4y - 2\].

Step 3: Final expression

The expression has been simplified by distributing and combining like terms. The final simplified expression is \[4y - 2\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term by a group of terms inside a parenthesis. This helps to "distribute" or spread the multiplication across each term. In simpler words, whenever you see a number or variable outside the parenthesis, it needs to be multiplied with every single term inside the parenthesis.

• For example, in the expression \( -2(1 + 7y) \), you distribute by multiplying \(-2\) by \(1\) and \(7y\).
• This results in \(-2 \, \times \, 1 = -2\) and \(-2 \, \times \, 7y = -14y\).

Using the distributive property helps break down complex equations into simpler parts, making them easier to work with and understand. So the expression \(18y - 2(1 + 7y)\) transforms to \(18y - 2 - 14y\) after using the distributive property effectively.

Combining Like Terms

Combining like terms is another essential element of simplifying algebraic expressions. Like terms are terms that have the same variable(s) raised to the same power. In the expression \(18y - 2 - 14y\), the terms \(18y\) and \(-14y\) are like terms because both contain the variable \(y\).
To combine like terms:

• First, identify all terms in the expression that contain the same variable or are constants.
• Then, perform the necessary arithmetic operation (addition or subtraction) on these terms.

In the given example: \(18y - 14y\). Subtract \(14y\) from \(18y\) resulting in \(4y\).
This step helps streamline the expression by reducing the number of terms, leading us to a more simplified expression of \(4y - 2\). This makes further calculations easier and the expression easier to understand.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition or multiplication. They form the basis for a lot of algebraic operations and problem-solving.
In mathematics, expressions don't have an "equals" sign unless they are equations, so an expression like \(18y - 2(1 + 7y)\) merely describes a value for a given \(y\).
Understanding components of algebraic expressions involves:

• Identifying terms: Each part of the expression separated by a "+" or "-" sign.
• Recognizing constants: Numbers that stand alone, such as \(-2\) in our expression.
• Understanding variables: Letters that represent unknown values, in this case, \(y\).

Simplifying algebraic expressions, as in our example, leads to easier interpretation or solving for specific variables. You learn to systematically apply mathematical rules to transform expressions to their simplest form, a valuable skill for solving equations.