Problem 86 - Algebra A Combined Function

Question

Match each expression on the left with its simplification on the right. Not all letters on the right must be used and a letter may be used more than once. a. \(3 y\) b. \(9 y-6 y^{2}\) c. \(10 x\) d. \(25 x^{2}\) e. \(10 x-6\) f. none of these $$ (15 x-3)-(5 x-3) $$

Step-by-Step Solution

Answer

The expression matches with c: 10x.

Step 1: Simplify the Expression

Let's simplify the given expression: \( (15x - 3) - (5x - 3) \). Start by distributing the negative sign inside the parentheses: \(15x - 3 - 5x + 3\).

Step 2: Combine Like Terms

Combine the like terms: \(15x - 5x + (-3 + 3)\). This simplifies to \(10x + 0\).

Step 3: Identify the Simplification

The expression \(10x\) from Step 2 corresponds with expression c., which is \(10x\). Therefore, the simplification of the original expression is \(10x\), matching it with option c.

Key Concepts

Combining Like Terms

In algebra, the ability to combine like terms is a fundamental skill for simplifying expressions. Like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients. For example, in the expression \(15x - 5x\), both terms are like terms because they consist of the same variable \(x\). This allows us to subtract the coefficients, simplifying it to \(10x\).
When simplifying expressions, always look for terms with identical variable parts. Once identified, sum up or subtract the numerical values (known as coefficients) directly. This process can significantly reduce the complexity of an expression and make further calculations easier.
Remember, combining like terms is like organizing similar items together, it makes the expression balanced and much simpler to work with.

Distributive Property

The distributive property is a useful tool in algebra that allows us to distribute a factor outside of parentheses to each term within. It is formally stated as \(a(b + c) = ab + ac\). This property is particularly handy when simplifying expressions that involve parentheses.
In our example, the expression \((15x - 3) - (5x - 3)\) uses the distributive property implicitly. By thinking of the negative sign in front of the second parenthesis as a \(-1\) being distributed, we effectively simplify the expression to \(15x - 3 - 5x + 3\).
The distributive property allows for easier handling of expressions, especially when solving equations. It provides a way to "open up" parentheses and transform an expression into a simpler form, paving the path for further simplification.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Unlike equations, these expressions do not have an equals sign. Simplifying algebraic expressions involves utilizing various mathematical concepts to make the expression easier to work with.
In the context of the given exercise, our expression is \((15x - 3) - (5x - 3)\). By applying rules like the distributive property and combining like terms, we transform this expression into its simplest form, \(10x\).
It's important to understand that the primary goal of working with algebraic expressions is to make them as simple as possible. Simplification makes it easier to see the relationships between terms and can assist in solving more complex algebraic problems.

Negative Sign Distribution

Distributing a negative sign across an expression is similar to distributing any other number or variable, but with a crucial twist: it changes the sign of each term within the parentheses. This can be visualized as multiplying each term inside by \(-1\).
For example, in the expression \((15x - 3) - (5x - 3)\), removing the parentheses involves changing the signs of each term in the second set. So, \(5x - 3\) becomes \(-5x + 3\) upon distribution of the negative sign.
This technique is key in simplifying expressions and preparing them for further operations like combining like terms. Recognizing and correctly applying negative sign distribution is essential for properly simplifying algebraic expressions, leading to correct and concise results.