Problem 71 - Elementary Algebra

Question

Find the sums for the the following problems. \([2+(-4)]+[17+(-19)]\)

Step-by-Step Solution

Answer

Answer: -4

Step 1: Identify the expressions inside the parentheses

We have two sets of parentheses in this expression: \(([2+(-4)])\) and \(([17+(-19)])\). Our first task will be to simplify both of these expressions.

Step 2: Simplify expressions inside the parentheses

\(2 + (-4) = -2\) and \(17 + (-19) = -2\). Now that we've simplified inside, our expression looks like this: \((-2) + (-2)\).

Step 3: Simplify the remaining expression

Now we only have one operation left to do, and that is to sum both the simplified expressions from the previous step: \((-2) + (-2) = -4\) The final answer is \(-4\).

Key Concepts

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the manipulation and solution of algebraic expressions and equations. It is where students first encounter symbols and letters used to represent numbers and quantities in formulas. Simplifying algebraic expressions, as in the exercise \( [2+(-4)]+[17+(-19)] \), is a fundamental skill in this discipline.

When faced with algebraic expressions, the first step is often to identify and simplify the terms within parentheses. This process, which involves combining like terms and applying the distributive property, sets the stage for further simplifications or solving for variables. It's important to get comfortable with this early on, as more complex algebra, including functions and polynomial equations, builds upon these elementary concepts.

Algebraic Addition

Algebraic addition involves combining algebraic expressions to form a single expression. In the context of the exercise, we see two sets of numbers being added in separate parentheses, and later, the resulting sums are added together. The key rule to remember is that when you add a negative number, it is equivalent to subtracting its absolute value.

For instance, in the expression \(2 + (-4)\), adding -4 is the same as subtracting 4 from 2. This understanding streamlines the simplification process: \(2 - 4 = -2\), and similarly for \(17 + (-19) = -2\). The next step is to address the outer addition: simplifying \( (-2) + (-2)\) becomes straightforward once we apply this logic of algebraic addition, where we simply keep adding the values, keeping track of the sign of each number.

Negative Numbers

Negative numbers are an essential part of algebra and represent quantities that are less than zero. They play a critical role in defining the solution sets of algebraic equations and expressions. In our example, after simplifying within the parentheses, we encounter the expression \( (-2) + (-2) \), which requires adding two negative numbers together.

A helpful tip for dealing with negative numbers is to think of them as debts or losses. If you lose 2 dollars and then lose another 2 dollars, your total loss is 4 dollars, which we represent as -4 in mathematics. This visualization helps to clarify why \( (-2) + (-2) = -4 \): you are combining the amounts of 'loss', leading to a larger 'loss' or negative amount. Understanding the handling of negative numbers is vital for successfully simplifying algebraic expressions as well as for more complex operations encountered later in algebra.