Problem 69 - Elementary Algebra

Question

Simplify. $$ \text { L. } 5 x-2(4 x-5) $$

Step-by-Step Solution

Answer

The simplified expression is \(-3x + 10\).

Step 1: Distribute the Parentheses

Apply the distributive property to the expression inside the parentheses. This means multiplying each term inside the parentheses by the factor outside: \[ -2(4x - 5) \]This becomes:\[ -2 imes 4x + -2 imes (-5) \]Simplifying further gives:\[ -8x + 10 \]

Step 2: Combine Like Terms

Now, combine the terms from the result of the distribution with the remaining terms from the original expression:\[ 5x - 8x + 10 \]Combine the x terms:\[ (5x - 8x) + 10 \]Simplifying the x terms gives:\[ -3x + 10 \]

Step 3: Simplified Expression

The expression is simplified and no further like terms exist, so the final simplified expression is:\[ -3x + 10 \]

Key Concepts

Distributive Property

The distributive property is an important concept in algebra that helps in breaking down expressions to simplify them. It states that for any numbers or variables, the formula is:

• \( a(b + c) = ab + ac \)

This means that when you have a term outside a set of parentheses, you "distribute" it to each term inside. In the given exercise, we need to distribute \(-2\) over the terms \(4x - 5\). This requires multiplying \(-2\) by both \(4x\) and \(-5\):

• \(-2 \times 4x = -8x\)
• \(-2 \times (-5) = 10\)

By following this property, we transform the expression \(-2(4x - 5)\) into \(-8x + 10\). Each step in distribution should be done carefully, and it's always useful to double-check your work to ensure accuracy.

Combining Like Terms

Once you have distributed the terms correctly, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same powers. In our original expression, after using the distributive property, we have:

• \(5x\)
• \(-8x\)
• \(+10\)

The terms \(5x\) and \(-8x\) are like terms because they both contain the variable \(x\). To combine them, you simply add or subtract their coefficients:

• \(5x - 8x = -3x\)

The constant term \(+10\) remains unchanged since it does not have a like term in this particular expression. Combining like terms helps to simplify the expression and makes it easier to evaluate if needed.

Simplifying Expressions

Simplifying expressions involves taking an algebraic expression and making it as compact and uncomplicated as possible. The ultimate goal is to combine all possible like terms and to perform any available arithmetic operations. We've already used the distributive property and combined like terms to move towards this goal.After combining like terms in the given exercise, we simplified the expression to \(-3x + 10\). This means there are no more operations that can be performed. The expression is now in its simplest form.Keys to simplifying expressions include:

• Using the distributive property carefully for any terms in parentheses.
• Identifying and combining all like terms.
• Making sure that each variable in the expression is grouped and simplified as much as possible.

Simplifying expressions makes them easier to work with and ensures clearer understanding of the relationships between variables. Remember, practice is key to becoming proficient at these steps!