Problem 53 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$4 y+(-13 z)+(-10 y)+17 z$$

Step-by-Step Solution

Answer

The simplified form of the algebraic expression is \(-6y + 4z\).

Step 1: Identifying Like Terms

In the given expression, \(4y + (-13z) + (-10y) + 17z\), the like terms are, terms of y which are \(4y\) and \(-10y\), and terms of z which are \(-13z\) and \(17z\).

Step 2: Combine Like Terms

To simplify the expression, add or subtract the coefficients of the like terms. Combine the 'y' terms together: \(4y - 10y = -6y\) and the 'z' terms together: \(-13z + 17z = 4z\).

Step 3: Write the Simplified Expression

After combining like terms, write them as a simplified expression, so the final result is \(-6y + 4z\)

Key Concepts

Like Terms

To fully grasp algebraic expressions, it's crucial to understand the concept of "like terms." In algebra, like terms are terms that have the same variables raised to the same powers. Essentially, they "look" similar except for their coefficients, which are the numbers multiplying the variables. Identifying like terms is the first go-to step when you're trying to simplify algebraic expressions.
For instance, in the expression given,

• the terms involving the variable "y" are like terms: \(4y\) and \(-10y\)
• those involving "z" are also like terms: \(-13z\) and \(17z\).

Recognizing these similarities allows you to group and eventually simplify expressions. Once like terms are identified, you can move on to combining them, which is the next step in simplifying.

Simplifying Expressions

Simplifying expressions is about making the expression more manageable by reducing it into the simplest form. Think of it like cleaning up a messy room—everything needs to be organized and put where it belongs. In math, simplifying means rearranging terms to reduce complexity, usually by combining like terms.
When given an expression like \(4y + (-13z) + (-10y) + 17z\), you start by looking for like terms.

• Notice that \(4y\) and \(-10y\) can be combined.
• Similarly, \(-13z\) and \(17z\) can be combined.

The primary goal in simplifying is to shrink the expression, making it as straightforward and elegant as possible. You'll end up with a cleaner, more concise expression, with each distinct set of variables reduced to a single term.

Combining Like Terms

Combining like terms is a key process when simplifying algebraic expressions. Once like terms are identified, you combine them by adding or subtracting their coefficients. This step trims the fat from the expression, keeping just what's necessary.
Take the "y" terms from our example:

• Combine them as \(4y - 10y\), which simplifies to \(-6y\).

Next, look at the "z" terms:

• Combine them as \(-13z + 17z\), which simplifies to \(4z\).

When combined, the algebraic expression becomes \(-6y + 4z\). This practice of merging like terms not only simplifies the expression but also makes it more intuitive and easier to interpret or solve in future problems. Knowing this can unlock a whole new level of comfort and proficiency in algebra, preparing you for more complex challenges.