Problem 48 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

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

Step 1: Expand the Expression

Distribute the ",-2\" across the terms in the parentheses. This means multiplying ",-2\" by each term inside the parentheses:\[-2(1 + 7y) = (-2)\cdot 1 + (-2)\cdot (7y) = -2 - 14y\]

Step 2: Combine Like Terms

Substitute the expanded form of ",-2(1 + 7y)\" into the original expression and combine like terms:\[18y - 2 - 14y\]Now combine the terms with \"y\":\[18y - 14y = 4y\]Next, put the constant term:\[4y - 2\]

Key Concepts

Distributive Property

The distributive property is a useful algebraic principle that allows us to simplify expressions where a term is being multiplied by a sum (or difference) inside parentheses. In expressions like this, you can multiply the term outside the parentheses by each term inside. This helps break down the problem into smaller, more manageable parts.

For example, in the expression \(-2(1 + 7y)\), we apply the distributive property by multiplying \(-2\) with each term inside the parentheses:

• Multiply \(-2\) by \(1\): \(-2 imes 1 = -2\)
• Multiply \(-2\) by \(7y\): \(-2 imes 7y = -14y\)

This yields \(-2 - 14y\). Now, the expression is simpler to work with, making it easier to continue solving or simplifying further.

Using the distributive property effectively helps lay the groundwork for correctly simplifying algebraic expressions and solving equations.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For example, \(18y\) and \(-14y\) are like terms because they share the same variable \(y\).

To combine like terms, you simply add or subtract their coefficients. In the given expression, \(18y - 14y\), both terms contain the variable \(y\). We combine them by subtracting their coefficients:

• Take \(18y\) and subtract \(-14y\) from it: \(18y - 14y = 4y\)

After combining, you end up with \(4y\). This process simplifies the expression, making it more concise and easier to work with.

Remember, the key to combining like terms is to look for terms that share the same variable and degree. This helps maintain the accuracy of the expression while reducing its complexity.

Linear Expressions

Linear expressions are algebraic expressions where each term is either a constant or the product of a constant and a single variable raised to the first power. They have a straightforward structure, often leading to equations that form straight lines when graphed.

For instance, the expression \(4y - 2\) is linear because it includes the term \(4y\), where \(y\) is to the first power, and the standalone constant \(-2\).

Characteristics of linear expressions include:

• They do not contain exponents higher than one on the variable.
• They only involve one variable or combinations that don't exceed the first degree.
• They are free of terms like \(y^2\), \(y/z\), or \(rac{1}{y}\).

Linear expressions are foundational in algebra because they form the simplest type of functions and equations, making them easier to understand and solve. They help students get comfortable with algebraic concepts before moving on to more complex quadratic or polynomial expressions.