Problem 39 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ x x-5-2 x-3-5(x-3) \times 2-8 x+15 $$

Step-by-Step Solution

Answer

The simplified expression is \(-2x + 17\).

Step 1: Distribute and Expand

Start by distributing the multiplication across the terms. In the expression \(-5(x-3)\), distribute \(-5\) to \(x\) and to \(-3\):\(-5(x-3) = -5x + 15\).Similarly, distribute \(2\) in the expression \(+2(x-3) = +2x - 6\).Substitute these into the original expression to get\(x - 5 - 5x + 15 + 2x - 8 + 15\).

Step 2: Combine Like Terms

Group similar terms together and simplify:- Combine the variable terms: \(x - 5x + 2x = (1-5+2)x = -2x\).- Combine the constant terms: \(-5 + 15 - 8 + 15 = 17\).This gives a simplified expression:\(-2x + 17\).

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental rule in algebra that helps us simplify expressions involving multiplication over addition or subtraction. It allows you to multiply a single term by each term inside a parenthesis.
This property states that for any numbers or expressions \(a, b,\ and\ c\),

• \( a(b + c) = ab + ac \)
• \( a(b - c) = ab - ac \)

Using the distributive property, we ensure that each term inside the parentheses is multiplied by the factor outside.
This step is crucial in problems like the one above where you encounter expressions such as

• \(-5(x-3)\)
• \(+2(x-3)\)

For \(-5(x-3)\), you distribute \(-5\) across both terms inside, leading to \(-5x + 15\).
Similarly with \(+2(x-3)\): distribute \(+2\) resulting in \(+2x - 6\).
Distributing correctly prevents errors in your calculations and ensures accurate manipulation of expressions.

The Art of Combining Like Terms

Combining like terms is another key concept in simplifying algebraic expressions. It involves merging terms that have identical variable parts.
Terms are considered 'like' if they have the same variable raised to the same power. For example,

• \(x\), \(-x\), and \(2x\) are like terms.
• Constants, such as \(-5, 15,\ and\ -8\), can also be combined as like terms.

In the given exercise, once we had the expression like \(x - 5x + 2x\), these are all like terms because they all involve \(x\).
By summing their coefficients, we obtained \((1-5+2)x = -2x\).
The constant terms were equally added: \(-5 + 15 - 8 + 15 = 17\).
Properly combining like terms leads to simpler forms and helps in understanding the expression better.

Mastering Algebraic Expression Manipulation

Algebraic expression manipulation involves several techniques like distributing terms and combining like terms to transform expressions into simpler or more useful forms.
This isn't just about solving problems, but also about enhancing comprehension of how algebra behaves under various operations.
When tackling a problem involving an expression, you may need to:

• Identify operations, such as addition, subtraction, multiplication, within expressions.
• Use the distributive property to remove parentheses.
• Rearrange terms to place like terms together.
• Combine like terms using basic arithmetic operations.

Practicing these techniques is essential for developing strong algebraic skills.
In any expression, remember your goal is to simplify it into the clearest form, enabling more straightforward problem-solving or deeper analysis.
Efficient manipulation aids in a smooth transition towards more complex algebraic challenges.