Problem 100 - Elementary Algebra

Question

Simplify: \(-3(2 x-11)-4(5 x-8)\)

Step-by-Step Solution

Answer

-26x + 65

Step 1: Distribute the constants

First, distribute \(-3\) through \(2x - 11\) and \(-4\) through \(5x - 8\). This means applying the distributive property: \(-3(2x - 11) = -6x + 33\) and \(-4(5x - 8) = -20x + 32\).

Step 2: Combine like terms

Now, combine the like terms from the results of the distribution. You have \(-6x + 33\) and \(-20x + 32\). Combining these, you get \(-6x - 20x + 33 + 32\).

Step 3: Simplify the expression

Add the like terms together. Combine the \(-6x\) and \(-20x\) to get \(-26x\). Combine the \(+33\) and \(+32\) to get \(+65\). Thus, the simplified expression is \(-26x + 65\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions. It states that you can distribute a multiplied value across terms inside a parenthesis. For example, when you have \(-3(2x - 11)\), you multiply \(-3\) by both \(2x\) and \(-11\). So, it becomes \(-3 \cdot 2x - 3 \cdot (-11) = -6x + 33\). In our specific problem, we use the distributive property twice: once for \(-3(2x - 11)\) and once for \(-4(5x - 8)\). This gives us \(-6x + 33\) and \(-20x + 32\). Thoroughly applying the distributive property is the key first step to simplifying algebraic expressions.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are those that have the same variable raised to the same power. In the expression resulting from the distribution step \(-6x + 33\) and \(-20x + 32\), the \(-6x\) and \(-20x\) are like terms because both are terms with \(x\). Similarly, \(33\) and \(32\) are like terms because they are constant numbers. You combine \(-6x\) and \(-20x\) to get \(-26x\), and you combine \(+33\) and \(+32\) to get \(+65\). Therefore, combining like terms helps simplify the expression further to a single, more manageable form.

Simplify Algebraic Expressions

Simplifying algebraic expressions involves applying various steps to make the expression as simple as possible. First, you use the distributive property to remove parentheses. Then, you combine like terms to reduce the number of terms. Through these steps, the expression becomes easier to work with and understand. In our example, we took \(-3(2x - 11) - 4(5x - 8)\) and simplified it down to \(-26x + 65\). Simplification makes it clear that the expression represents the same quantity but in a more streamlined and comprehensible form.