Problem 56 - Algebra 1: Concepts and Skills

Question

Which expression is simplified? $$(F) 7+5k-5k$$ $$ (G)3 x-9+2 x^{2} $$ $$ (H)-8 g+5-8 g $$ $$ (J)11 z-4 z $$

Step-by-Step Solution

Answer

After going through all of the given expressions, it is clear that Expression (F) is the simplified one as it only contains the number 7.

Step 1: Simplify Expression (F)

Let's start by simplifying expression (F), which is \(7+5k-5k\). Combine the like terms, in this case \(5k\) and \(-5k\), and we get \(7+0\). That simplifies to \(7\).

Step 2: Simplify Expression (G)

Expression (G) is given as \(3x-9+2x^2\), and there are no like terms to combine, so it remains as \(3x + 2x^2 - 9\). We rearrange terms as it is preferred in mathematics to have powers of a variable arrange in descending order.

Step 3: Simplify Expression (H)

Simplify expression (H), written as \(-8g+5-8g\). Combine the like terms, in this case \(-8g\) and \(-8g\), and we get \(5-16g\). We rearrange into \(-16g+5\) as it's prefered to have terms with variables first.

Step 4: Simplify Expression (J)

Expression (J) is \(11z-4z\). Combine the like terms, which are \(11z\) and \(-4z\), to get \(7z\) in this case.

Key Concepts

Combining Like Terms

When working with algebraic expressions, one of the most fundamental skills is combining like terms. This process involves adding or subtracting terms that have the same variable raised to the same power. Like terms are algebraic expressions that share the same variable and exponent. For example, in the expression 7+5k-5k, the terms 5k and -5k are like terms because they both contain the variable k to the first power.

To combine these like terms, we simply perform the arithmetic operation indicated; in this case, we add 5k (which is the same as +5k) and -5k. They cancel each other out, leaving us with 7, which is the simplified form of the expression. Likewise, in the expression 11z-4z, by subtracting 4z from 11z, we get 7z. Combining like terms makes an expression more manageable and sets the stage for further algebraic manipulation.

Here are some key points to remember when combining like terms:

Variable Expressions

Variable expressions are mathematical phrases that contain numbers, variables (like x, y, or z), and operation signs. Variables represent unknown quantities and are a core component of algebra. They allow us to write general rules and formulas that apply to many different situations. A variable expression becomes particularly useful when we work with equations and inequalities, giving us the flexibility to manipulate and solve for the unknowns.

In the provided exercise, each option represents a different variable expression. For instance, 3x-9+2x^2 is a variable expression with different terms involving the variable x. When dealing with variable expressions, always remember to:

Algebraic Simplification

Algebraic simplification is the process of reducing an algebraic expression to its simplest form. This makes the expression easier to understand and work with. Simplifying an expression doesn't change its value; rather, it makes it more straightforward. This is achieved by performing a series of algebraic operations, such as combining like terms, using distribution laws, and canceling terms where possible.

For example, the expression -8g+5-8g can be simplified by combining the like terms -8g and -8g to get -16g and then rewriting the simplified expression as -16g+5. Simplified algebraic expressions are particularly helpful when solving equations because they can make complex problems more accessible.

Algebraic simplification involves understanding the properties of numbers and operations and applying these strategically to rewrite expressions in a more usable format. Here are some essential aspects of algebraic simplification: