Problem 45
Question
For the following exercises, simplify the expression. $$ 9(y+8)-27 $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(9y + 45\).
1Step 1: Distribute the 9
Start by distributing the 9 to both terms inside the parentheses. This means multiplying 9 by each term: \(9 \times y\) and \(9 \times 8\). This results in \(9y + 72\).
2Step 2: Subtract 27 from the Expression
Now, take the expression from Step 1 and subtract 27 from it: \(9y + 72 - 27\).
3Step 3: Combine Like Terms
Simplify the expression by combining the numbers \(72\) and \(-27\). Calculate \(72 - 27\) which gives you 45. Thus, the expression becomes \(9y + 45\).
Key Concepts
The Distributive PropertyCombining Like TermsUnderstanding Algebraic Expressions
The Distributive Property
The distributive property is a fundamental concept in algebra that helps simplify expressions. It allows you to distribute a number multiplied by a sum or difference inside parentheses to each term independently.
For example, in the expression \(9(y + 8)\), the distributive property lets us multiply 9 by both \(y\) and 8. This results in two separate products: \(9 \times y\) and \(9 \times 8\), which simplifies to \(9y + 72\).
This property makes it easier to work with expressions that involve parentheses, leading to simplification and making calculations more straightforward. Remember, the key is to multiply the number outside the parentheses by each term inside, one by one.
For example, in the expression \(9(y + 8)\), the distributive property lets us multiply 9 by both \(y\) and 8. This results in two separate products: \(9 \times y\) and \(9 \times 8\), which simplifies to \(9y + 72\).
This property makes it easier to work with expressions that involve parentheses, leading to simplification and making calculations more straightforward. Remember, the key is to multiply the number outside the parentheses by each term inside, one by one.
Combining Like Terms
Combining like terms is another essential skill to simplify algebraic expressions effectively. Like terms are terms that share the same variable raised to the same power. For instance, in the expression \(9y + 72 - 27\), there are constant terms 72 and -27 that are considered like terms.
To combine them, you add or subtract the coefficients. In this case, we subtract 27 from 72. The calculation \(72 - 27\) gives 45.
Thus, the expression simplifies to \(9y + 45\). This process of combining like terms helps reduce the complexity of algebraic expressions and prepares them for other operations or equations.
To combine them, you add or subtract the coefficients. In this case, we subtract 27 from 72. The calculation \(72 - 27\) gives 45.
Thus, the expression simplifies to \(9y + 45\). This process of combining like terms helps reduce the complexity of algebraic expressions and prepares them for other operations or equations.
Understanding Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are used to represent mathematical computations in a general form. In our exercise, the expression \(9(y+8)-27\) is an example of an algebraic expression that includes both variables and constants.
- Variables: These are symbols, like \(y\), representing unknown values that can change or vary.
- Constants: These are fixed numbers, such as 9, 8, and 27 in the expression.
- Operations: These include addition, subtraction, multiplication, and division.
Other exercises in this chapter
Problem 45
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