Problem 104 - Intermediate Algebra

Question

Simplify each expression. $$2.1(4+5 z)+0.9 z$$

Step-by-Step Solution

Answer

The simplified expression is \(8.4 + 11.4z\).

Step 1: Distribute the 2.1

Begin by distributing the 2.1 into each term inside the parentheses. The expression is: \[ 2.1 \times 4 + 2.1 \times 5z \]Calculate each part:\[ 2.1 \times 4 = 8.4 \]\[ 2.1 \times 5z = 10.5z \]So after distributing, the expression becomes:\[ 8.4 + 10.5z + 0.9z \]

Step 2: Combine like terms

Now, you combine the like terms in the expression. The terms with \(z\) are \(10.5z\) and \(0.9z\). Add these together:\[ 10.5z + 0.9z = 11.4z \]So the expression now simplifies to:\[ 8.4 + 11.4z \]

Step 3: Final Expression

Now you have the fully simplified expression. It cannot be simplified further as there are no other like terms:\[ 8.4 + 11.4z \]

Key Concepts

The Distributive Property

One of the foundational concepts in algebra, the distributive property helps us to efficiently simplify expressions involving multiplication over addition. Imagine you have a group of terms inside parentheses, such as \(4 + 5z\). When a number, like 2.1, is outside the parentheses and needs to be multiplied with each term inside, this is a perfect situation to use the distributive property.The distributive property states: \[ a(b+c) = ab + ac \]This means you multiply the outside number with each term inside and add them up:

• For \(2.1(4 + 5z)\), we do: \(2.1 \times 4 + 2.1 \times 5z\)
• This simplifies to: \(8.4 + 10.5z\)

This step simplifies the expression by breaking it down into easier parts that can be individually calculated.

Combining Like Terms

The process of combining like terms is all about simplification by putting terms together that share the same variable. In our example, once we have distributed and simplified the terms inside the parentheses, the expression becomes \(8.4 + 10.5z + 0.9z\).The concept of 'like terms' involves looking for terms that have identical variables raised to the same power:

• Here, both \(10.5z\) and \(0.9z\) have the variable \(z\), so they are like terms.
• Adding these together, you get: \(10.5z + 0.9z = 11.4z\)

Combining like terms makes expressions concise and easier to manage, resulting in a simplified expression \(8.4 + 11.4z\). This is key to mastering algebraic operations.

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operation symbols. They are fundamental in algebra and allow us to express relationships and solve problems.An expression such as \(2.1(4+5z) + 0.9z\) includes all these components:- **Terms**: These are parts of the expression separated by plus or minus signs, like \(2.1(4+5z)\) and \(0.9z\).- **Coefficients**: These are numbers that multiply the variables. For instance, in \(5z\), 5 is the coefficient.- **Variables**: These are symbols that stand in for numbers we don't yet know. The \(z\) in our example is a variable.Algebraic expressions are the building blocks of algebra, enabling us to represent problems abstractly and solve them through systematic simplification.