Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the expression. \(9(y+8)-27\)

Step-by-Step Solution

Answer

The simplified expression is \(9y + 45\).

Step 1: Distribute the 9

Apply the distributive property to the expression. Multiply the 9 across the terms inside the parentheses: \(9(y + 8) - 27\) becomes \(9y + 72 - 27\).

Step 2: Combine Like Terms

Combine the constant terms in the expression. Add or subtract these constants to simplify: \(9y + 72 - 27\) simplifies to \(9y + 45\).

Key Concepts

Distributive Property

When simplifying an expression using the distributive property, you are essentially multiplying a single term by each of the terms inside a parenthesis.
This is fundamental in algebra and helps in rewriting expressions without parentheses.
In the exercise given, the distributive property is applied by multiplying the number 9 with each term inside the parentheses, which are the 'y' and the 8.
Here's how it works:

• Multiply 9 and y to get 9y
• Multiply 9 and 8 to get 72

By applying the distributive property, we transform the expression from \(9(y+8)\) to \(9y + 72\). This step is crucial as it simplifies the process of further calculations by getting rid of the parentheses.
Remember, the distributive property follows the form: \(a(b + c) = ab + ac\).Understanding this step is key, as it lays the foundation for simplifying more complex expressions.

Combining Like Terms

Combining like terms is a method used to simplify expressions further by summing terms that have the same variable component.
In the context of the linear expression from our exercise, once the distributive property has been applied resulting in \(9y + 72 - 27\), the next step is to focus on rearranging the expression by grouping similar terms.
Here's how you can spot like terms:

• Look for terms that have the same variable and exponent. In our case, this is the term with a 'y', which is 9y.
• Consider constant terms, which are numbers without variables, like 72 and 27 here.

Once identified, we combine them by simply performing the arithmetic operation - in this instance, subtracting 27 from 72, which results in 45.
Thus, the expression transforms from \(9y + 72 - 27\) to \(9y + 45\).
By combining like terms, the expression becomes more concise, making it easier to understand and solve.

Linear Expressions

Linear expressions are algebraic expressions where the variable is raised only to the power of one.
They do not include variables with exponents greater than one or any products of variables.
This simplicity makes them crucial to master early on when learning algebra.A linear expression is typically in the form of \(ax + b\).
In our exercise, the final simplified expression \(9y + 45\) represents a linear expression, where 9 is the coefficient of the variable 'y', and 45 is the constant.

• The term '9y' denotes that for any given value of 'y', you would multiply it by 9.
• The term '45' is simply added to the outcome of '9y'.

Linear expressions illustrate how changes in the variable can affect the entire expression's value. They are fundamental in building equations and functions.
Understanding their properties will serve you well in dealing with more advanced algebraic concepts and in applying math to real-world scenarios like computing costs and rates.