Problem 62 - College Algebra

Question

simplify each algebraic expression. $$ 2(5 x-1)+14 x $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(24x - 2\).

Step 1: Expanding the Brackets

The first step is to expand the brackets in the algebraic expression. This leads us to the equation: \(2 * 5x - 2 * 1 + 14x\). When we multiply these, we get \(10x - 2 + 14x\).

Step 2: Combining Like Terms

Next, we combine the like terms, which are \(10x\) and \(14x\). This results in \(24x - 2\) when we add these together.

Step 3: Final Simplification

With all like terms combined, the expression is now in its simplest form, which is \(24x - 2\).

Key Concepts

Expanding Brackets

Expanding brackets is a foundational skill in algebra that involves distributing a coefficient across terms inside a bracket. In our exercise, we need to expand the expression: \[ 2(5x - 1) + 14x \] The first part involves the brackets \(2(5x - 1)\). To expand, you distribute the 2 across each term inside the brackets.

• Multiply 2 by 5x, giving you \(10x\).
• Then multiply 2 by -1, resulting in \(-2\).

After expanding, the terms inside the bracket become \(10x\) and \(-2\). The expanded expression is: \[ 10x - 2 + 14x \] Expanding brackets helps simplify the algebraic expression and prepares it for the next steps of simplification.

Combining Like Terms

Once you've expanded the brackets, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, the like terms are the \(10x\) and \(14x\) because they both have the variable \(x\). Combining like terms involves adding or subtracting these terms. Here's how to do it:

• Take the x terms: \(10x\) and \(14x\).
• Add them together: \(10x + 14x\).
• This results in \(24x\).

After combining like terms, the expression is simpler and can now be written as: \[ 24x - 2 \] This step is crucial for reducing the complexity of expressions and getting closer to the simplest form.

Simplification Steps

The final step in simplifying an algebraic expression is to ensure that all like terms are combined and the expression is as straightforward as possible. By now, you have already expanded and combined like terms, resulting in the expression: \[ 24x - 2 \] This expression is now in its simplest form because:

• There are no more like terms to combine.
• It consists of a single variable term \(24x\) and a constant \(-2\).

At this stage, no further simplification is possible. Simplifying expressions to their smallest possible form makes them much easier to work with, especially when solving more complex equations or algebraic problems.