Problem 93 - Elementary Algebra

Question

Problem: Simplify: \(8(4 x-1)-3(2 x-5)\) $$ \text { Incorrect Answer: } \begin{aligned} 8 &(4 x-1)-3(2 x-5) \\ &=8(4 x)+8(-1)-3(2 x)-3(5) \\ &=32 x-8-6 x-15 \\ &=26 x-23 \end{aligned} $$

Step-by-Step Solution

Answer

26x + 7

Step 1 - Distribute 8 inside the first parenthesis

Apply the distributive property to the expression inside the first parenthesis: \[8(4x-1) = 8 \times 4x + 8 \times (-1)\] This results in: \[32x - 8\]

Step 2 - Distribute -3 inside the second parenthesis

Similarly, apply the distributive property to the expression inside the second parenthesis: \[-3(2x-5) = -3 \times 2x + (-3) \times (-5)\] This results in: \[-6x + 15\]

Step 3 - Combine the distributed terms

Combine the results from Step 1 and Step 2: \[32x - 8 - 6x + 15\]

Step 4 - Simplify the expression

Combine like terms to simplify: \[32x - 6x = 26x\] \[-8 + 15 = 7\] So, the simplified expression is: \[26x + 7\]

Key Concepts

Distributive Property

Understanding the distributive property is key to simplifying algebraic expressions. It is a useful tool that allows you to break down complex expressions into simpler parts. Essentially, you apply a single multiplier to each term inside a parenthesis. Consider the exercise:

Step 1: Distribute 8 into the first parenthesis:
Use the distributive property to change:
8(4x - 1) into 8 * 4x + 8 * (-1), which simplifies to 32x - 8.

Step 2: Similarly, distribute -3 into the second parenthesis:
Again, using the distributive property, change:
-3(2x - 5) into -3 * 2x + (-3) * (-5), which simplifies to -6x + 15.

Remember, the key idea here is that multiplying each term inside the parenthesis separately can simplify the expression. This property is fundamental in algebra and will be used frequently.

Combining Like Terms

To simplify an algebraic expression, it is often necessary to combine like terms. Like terms are terms that have the same variable raised to the same power. In our example:

Consider combining the expressions you obtained from the distributive property:
32x - 8 and -6x + 15.

Step 3: Combine these terms:
You can rewrite: 32x - 8 - 6x + 15.

The next step involves combining the like terms:
32x and -6x are like terms because they contain the same variable, 'x'. Similarly, -8 and 15 are constants and can be combined.

Combining like terms:
32x - 6x gives you 26x.
Similarly, combining -8 + 15 gives you 7.

Combining like terms helps consolidate the expression into simpler ones, making it easier to work with in later steps.

Simplification

Once you have used the distributive property and combined like terms, your expression is nearly simplified. Simplification makes an algebraic expression easier to read and work with. Let’s finalize our example:

Step 4: Combine the results:
So now we have: 32x - 8 - 6x + 15 simplifying to 26x + 7.

This is your final simplified expression.
Simplification reduces the expression to its most manageable form, which is especially useful for solving equations, graphing, or any further manipulation. The correct approach allows you to avoid errors and ensures that you get the right answers efficiently.

In brief, always remember to apply the distributive property first, then combine like terms, and finally check for further simplification for a clear and concise result.