Problem 50 - College Algebra

Question

For the following exercises, simplify the expression. $$ 8(3-m)+1(-8) $$

Step-by-Step Solution

Answer

The simplified expression is \(16 - 8m\).

Step 1: Distribute the Constants

To simplify the expression, first distribute the constants inside the parentheses. Multiply 8 by each term in the first parenthesis: 8(3) and 8(-m), separately. Also, distribute the 1 with the term in its parentheses: 1(-8). This gives us: 24 - 8m - 8.

Step 2: Combine Like Terms

Now that we have removed the parentheses through distribution, combine the like terms in the expression. The expression is: \(24 - 8m - 8\). The like terms here are the constant terms, 24 and -8. Combine them to get: \(16 - 8m\).

Step 3: Write the Final Simplified Expression

After simplifying by combining all like terms, the expression becomes \(16 - 8m\). This is the simplest form of the original given expression.

Key Concepts

Distributive Property

The Distributive Property is a fundamental idea in algebra that helps us simplify expressions. It allows us to multiply a number by a group of numbers added together. Think of it as sharing or distributing the number across the terms inside the parentheses.
When we have an expression like \[ 8(3 - m) \], we need to apply the distributive property. This means multiplying the 8 by each term inside the parentheses:

• First, multiply 8 by 3, which results in 24.
• Then, multiply 8 by \(-m\), resulting in \(-8m\).

These steps change our expression to \(24 - 8m\). This property not only helps simplify the expressions but is essential in making complex calculations more manageable.
By distributing the number outside the parentheses, we eliminate the parentheses, giving us a clear path to the next simplification steps.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that contain the same variable raised to the same power. In simpler terms, they "look alike" because they have the same variable components.
Consider the expression obtained after applying the distributive property: \(24 - 8m - 8\). Here, the like terms are the constant numbers 24 and \(-8\). To combine them, simply add or subtract their coefficients:

• Add 24 and \(-8\), which gives us 16.

Thus, the expression further simplifies to \(16 - 8m\).
Remember, when combining like terms, only the coefficients (numbers in front of the variables) are added or subtracted, while the variable part remains unchanged. This step is vital for reducing the complexity of algebraic expressions, helping us to arrive at the simplest form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (like addition or subtraction). They represent a mathematical phrase, much like a sentence in language.
In our example, the expression \(8(3 - m) + 1(-8)\) is an algebraic expression. It consists of:

• Constants (numbers) such as 8, 3, and -8.
• Variables like \(m\).
• Operations like addition and multiplication.

Algebraic expressions can look complicated at first, but by applying properties like distribution and combining like terms, we can simplify them easily.
This simplification helps us understand the relationships and constraints carried by the variables, making it easier to solve equations and understand mathematical patterns. Learning how to manage these expressions is foundational to algebra, forming the basis for solving more complex problems in math.