Problem 42 - Algebra A Combined Function

Question

Simplify the expressions. $$ (-2 x+5 y)+(2 x+11 y) $$

Step-by-Step Solution

Answer

The simplified expression is \(16y\).

Step 1: Identify Like Terms

First, identify the like terms in the expression. Like terms are those that have the same variable raised to the same power. In the expression \((-2x + 5y) + (2x + 11y)\), the like terms are \(-2x\) with \(+2x\) and \(+5y\) with \(+11y\).

Step 2: Combine Like Terms of x

Next, combine the like terms of \(x\). This involves adding the coefficients of the \(x\) terms together: \(-2x + 2x = 0x\). Here both terms cancel each other out, resulting in \(0x\).

Step 3: Combine Like Terms of y

Now, combine the like terms of \(y\) by adding their coefficients: \(5y + 11y = 16y\). This simplifies the terms of \(y\) to \(16y\).

Step 4: Write the Simplified Expression

Finally, write the simplified expression combining results from the previous steps: \(0x + 16y\). Since \(0x\) contributes nothing to the expression, it can be omitted, leaving \(16y\) as the simplified expression.

Key Concepts

Combining Like Terms

Combining like terms is a fundamental step in simplifying algebraic expressions. Like terms refer to terms that have the same variable components raised to identical powers. In the expression given, (-2x + 5y) + (2x + 11y), we notice terms that are considered like terms:

• -2x and +2x, both involving the variable x, and
• +5y and +11y, both involving the variable y.

To combine like terms, we add or subtract their coefficients — the numerical parts in front of the variables — while keeping the variables unchanged. For instance, the like terms -2x and +2x, with coefficients -2 and +2 respectively, cancel each other out to equal 0x, effectively removing the x term. Similarly, for the y terms, you just combine 5y and 11y by adding the coefficients, which simplifies to 16y. Understanding and identifying like terms make manipulating algebraic expressions much clearer.

Simplifying Expressions

Simplifying expressions involves combining like terms and making the expression easier to work with. The goal is to condense the expression down to its simplest form without changing its value. We start by looking for like terms, which are terms that have the same variables raised to the same powers. Once found, we can combine them by performing the arithmetic operations of addition or subtraction between their coefficients.

In our example, (-2x + 5y) + (2x + 11y), once you combine like terms, you end up with 0x + 16y. The term 0x is unnecessary because it equals zero, leaving us with the simplified result of 16y. Simplifying expressions is crucial in algebra because it allows you to better analyze and solve equations and inequalities. It helps in reducing complex algebraic problems to much easier ones that you can solve more efficiently.

Linear Expressions

Linear expressions are expressions where the variable is raised only to the first power, as seen in examples like ax + by + c. This makes them a key concept in algebra due to their straightforward nature, making them easy to manipulate and solve. An important aspect of linear expressions is their additive property, which allows for the combination of like terms.

In the expression (-2x + 5y) + (2x + 11y), both x and y are linear variables. It's worth noting how adding such terms leads to another linear expression, as seen in our result of 16y. Linear expressions are prevalent throughout much of algebra and form the foundation for understanding more complicated topics such as linear equations and inequalities. Learning to recognize and simplify linear expressions paves the way for mastering these advanced concepts in algebra.