Problem 65 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ 3 t[4-(t-3)]+t(t+5) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(17t - 2t^2\)

Step 1: Distributive property

The first step is to apply the distributive property to eliminate the brackets and parenthesis. Specifically, 3t is multiplied with both terms in the first bracket, and t is multiplied with both terms in the second parenthesis,\n resulting in \(12t - 3t^2 + t^2 + 5t\)

Step 2: Combine like terms

Next, the like terms—those with \(t\) and \(t^2\) are combined. This results in \(12t + 5t - 3t^2 + t^2\) being simplified to \(17t - 2t^2\)

Key Concepts

The Distributive Property

The distributive property is a fundamental principle in algebra, and it's essential for simplifying expressions. This property allows us to eliminate parentheses by distributing a multiplication over addition or subtraction. In essence, you multiply the term outside the parenthesis by each term inside it.

For instance, in the expression we have, the distributive property is applied as follows:

• The term \(3t\) is distributed over the terms in the brackets \([4 - (t - 3)]\).
• This means we multiply \(3t\) by both \(4\) and \(-(t - 3)\), which gives us \(3t \times 4 - 3t \times (t - 3)\).
• Similarly, \(t\) is multiplied by both terms in the second parenthesis \((t + 5)\).

So, you end up with new terms that can be combined further.

Practicing the distributive property helps in solving more complex algebraic expressions efficiently. You break down tasks into smaller parts, making the expressions much easier to handle.

Combining Like Terms

Once you've applied the distributive property and expanded the expression, the next step is to combine like terms. **Like terms** are terms that have the same variable raised to the same power. By combining these, you can simplify the expression to its most basic form.

In our example:

• We have two terms that contain \(t^2\): \(-3t^2\) and \(t^2\). When you combine them, their coefficients are simply added, resulting in \(-2t^2\).
• For the terms with \(t\), you have \(12t\) and \(5t\). Adding these like terms gives \(17t\).

So, by combining like terms, the expression \(12t - 3t^2 + t^2 + 5t\) simplifies to \(17t - 2t^2\). This process is crucial, as it makes complex expressions far simpler to understand and use.

Understanding Algebraic Expressions

Algebraic expressions can be a bit daunting initially, but understanding their components helps immensely. An **algebraic expression** is a mathematical phrase that can include numbers, variables, and operations without an equality sign. Let's demystify some components:

• **Terms**: These are separated by addition or subtraction, such as \(3t\) or \(-3t^2\).
• **Coefficients**: These are the numbers that multiply the variables, like \(3\) in \(3t\).
• **Variables**: The symbols, such as \(t\), represent numbers that can change or vary.
• **Constants**: Fixed numbers that stand alone without variables, though in our example, they don't appear explicitly.

Algebraic expressions are the building blocks of algebra. They enable us to represent real-world situations and solve equations intuitively. By understanding how to manipulate and simplify these expressions, you become proficient in tackling various algebra problems.