Problem 14 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 2(4 t-1)-4(1-t) $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(12t - 6\).

Step 1: Apply the distributive property

To simplify the given expression, we will first apply the distributive property. Multiply \(2\) with each term in the first parentheses and \(-4\) with each term in the second parentheses. This results in: \(2*4t - 2*1 - 4*1 + 4*t = 8t - 2 - 4 + 4t.\)

Step 2: Combine similar terms

Next, we can combine similar terms. Here, the similar terms are \(8t\) and \(4t\) (the terms with \(t\)), and \(-2\) and \(-4\) (constant terms). So, we add these similar terms together to get: \(8t + 4t -2 - 4 = 12t - 6.\)

Key Concepts

Distributive Property

The distributive property is a critical concept in algebra that allows you to simplify expressions by distributing a factor across terms within parentheses. In essence, you are 'distributing' the multiplication over addition or subtraction. Here's the formula for the distributive property:
\[ a(b + c) = ab + ac \
or
\a(b - c) = ab - ac \]
When we apply this property, as in the original exercise, we multiply 2 with each term inside the first parentheses, and similarly, -4 with each term inside the second parentheses. This step expands the expression, making it easier to combine like terms later on. This property is a powerful tool to simplify algebraic expressions, and understanding it conceptually is the key to unlocking many algebra problems.

Combining Like Terms

After using the distributive property, the next step in simplifying algebraic expressions is combining like terms. Like terms are terms that have the same variables raised to the same power. In other words, they 'look' the same. For example, \(3x\) and \(5x\) are like terms because they both have the single variable \(x\) to the power of 1.

When combining like terms, simply add or subtract the coefficients (the numerical parts) of these terms. In our exercise, \(8t\) and \(4t\) are combined to \(12t\), as they are both 't' terms. The constants \(-2\) and \(-4\) are also combined to give \(-6\). The process of combining like terms is essential for streamlining expressions and making them more manageable.

Algebraic Simplification

Algebraic simplification is the process of reducing an expression to its simplest form by performing operations like the distributive property and combining like terms, as well as others like factoring. The goal is to make the expression as concise and as straightforward as possible, often to ease further calculations or to make understanding the relationship between variables more clear.

In performing algebraic simplification, always look to apply these steps methodically. First, distribute to eliminate parentheses, as we did by multiplying 2 and -4 across their respective terms. Next, identify and combine like terms, which results in fewer terms and a cleaner expression. Lastly, if possible, factor common elements out of the terms to simplify the expression even further. Simplification is an efficient way to prepare algebraic expressions for solving equations or evaluating variable values.