Problem 53 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ -6(2-3 x)+10(5-x) $$

Step-by-Step Solution

Answer

The simplified expression is \(8x + 38\)

Step 1: Distribute Multiplication Over Addition/Subtraction

Distribute the -6 and +10 inside each grouping respectively: \[ -6*(2 - 3x) + 10*(5 - x) = -12 + 18x + 50 - 10x \]

Step 2: Combine Like Terms

Combine the constant values (-12 and 50) and the terms with x (18x and -10x) to simplify: \[ -12 + 18x + 50 - 10x = 18x - 10x - 12 + 50 = 8x + 38\]

Key Concepts

Removing Symbols of Grouping

When simplifying algebraic expressions, one of the first steps is often to remove symbols of grouping. These symbols include parentheses \( ( ) \), brackets \[ [ ] \], and braces \{ \} \}. The main goal here is to clear the way for combining like terms. Without the clutter of groupings, it becomes much easier to see which terms can be combined. For instance, in the exercise \( -6(2-3 x)+10(5-x) \), we start by distributing the multiplication across the terms within parentheses.

A piece of advice that can help you during this process is to carefully observe the signs before groups. A negative sign before the group means you multiply each term by -1. Simply put, when removing parentheses, reverse the sign of each term inside if they're preceded by a subtraction sign.

Combining Like Terms

Combining like terms is an essential step in simplifying algebraic expressions. Terms that are 'like' have the same variable part, raised to the same power. For example, \(2x\text{ and }5x\) are like terms because they both contain the variable \(x\) to the first power. When you combine like terms, you simply add or subtract their coefficients. So \(2x + 5x = 7x\).

In our textbook exercise, after distributing the multipliers, we could combine \(18x\) and \( -10x\) to get \(8x\), since they both have the same variable, \(x\), to the first power. Remember to pay close attention to the signs of the coefficients; they dictate whether you add or subtract the like terms. Being meticulous with this process will save you from making errors and lead to a correct simplification.

Distributive Property

The distributive property is a key rule in algebra that allows us to multiply a single term by each term within a grouping symbol. It states that \(a(b + c) = ab + ac\), regardless of what the terms are. It's crucial when you're dealing with expressions that have multiplication outside of grouping symbols.

In the exercise, we apply the distributive property: \( -6(2 - 3x) + 10(5 - x)\) becomes \( -12 + 18x + 50 - 10x\). Noting that we must multiply both terms within each set of parentheses by the number outside. Keep in mind, when a negative is distributed, it turns positives into negatives and vice versa. The distributive property is a powerhouse for simplifying because it breaks down complex, grouped terms into simpler, individual terms that can later be combined easily.