Problem 38 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 1 x+1 y-1 x-1 y+x-y $$

Step-by-Step Solution

Answer

Question: Simplify the given algebraic expression: $$ x + y - x - y + x - y $$ Answer: The simplified expression is $$ x - 2y $$.

Step 1: Identify like terms

In the given expression, we have the following terms: $$ 1x, 1y, -1x, -1y, x, -y $$ We can see that there are three types of terms: x terms, y terms, and negative x terms, and negative y terms.

Step 2: Combine like terms

Now that we have identified the like terms, let's combine them: $$ 1x + (-1x) + x $$ $$ 1y + (-1y) + (-y) $$

Step 3: Simplify the expression

Now let's simplify these terms: $$ (1 - 1 + 1)x $$ $$ (1 - 1 - 1)y $$ $$ 1x - 2y $$ So, the simplified expression is: $$ x - 2y $$

Key Concepts

Combining Like Terms

In algebra, combining like terms is one of the fundamental techniques used to simplify an expression. It's based on the principle that terms with the exact same variables can be added or subtracted from each other. This technique takes advantage of the distributive property, where terms with the same variable can be grouped as a single term.

Consider the expression from our exercise: \(1x + 1y - 1x - 1y + x - y\). Here, we can combine terms that contain the same variable. The 'x' terms (\(1x - 1x + x\)) can be combined, as can the 'y' terms (\(1y - 1y - y\)). By doing so, we simplify the expression into fewer terms which makes it easier to understand and work with in solving equations or inequalities.

Understanding how to combine like terms is crucial in algebra, as it aids in the simplification process and can make more complex algebraic manipulations more manageable.

Algebraic Expression Simplification

Simplification of an algebraic expression means to make it as simple as possible. This does not change the value of the expression; it only makes it easier to read or further manipulate. Simplification might include combining like terms, reducing fractions, or using exponent rules.

The expression from our problem, once like terms are combined, is reduced to \(x - 2y\). This is much simpler than the original, and there are no further steps we can take to simplify it. Simplification can help prevent errors in calculation and can be particularly useful before substituting numbers for variables.

In simplifying, we strive for an expression with the fewest possible terms, without any parentheses or like terms that could be combined, and with coefficients reduced to their lowest terms. This streamlined form of an expression makes it straightforward for students to plug in values or to use as the starting point for more advanced operations.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with solving equations and understanding how to manipulate algebraic expressions. It's the foundation on which more advanced areas of algebra, such as abstract algebra and algebraic geometry, are built. It typically covers operations with real numbers, basic forms of equations, and the functions of one variable.

The exercise presented focuses on a fundamental skill in elementary algebra: the ability to transform and simplify algebraic expressions. These skills are an essential step towards solving algebraic equations, which is a core aspect of this discipline. Proficiency in elementary algebra enables students to progress to more complex mathematical problems, including those in geometry, trigonometry, and calculus.

Elementary algebra is not just a set of rules to follow but a language through which we can describe and explore patterns and relationships in numbers. It's a toolkit that helps us model real-world situations and find solutions to everyday problems.