91Ó°ÊÓ

Algebraic expressions are the backbone of solving equations in algebra. They consist of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In the context of our exercise, the expression is set to capture a real-world scenario in mathematical terms. Here, the expression is simple yet powerful: the original quantity, denoted by the variable 'x', plus 85% of 'x'. Expressed algebraically, it's written as
\[ x + 0.85x \].
Understanding this expression is crucial because it encapsulates the entire problem. It tells us that we have an initial amount and are adding to it a fraction of itself, an idea seen frequently in many applications such as interest calculations and growth models.

Combining like terms is a fundamental process in simplifying algebraic expressions. Like terms are terms that have the exact same variable part, which means they have the same variables raised to the same powers. In our problem, we had two like terms: 'x' and '0.85x'. Both terms have the variable 'x' and can therefore be combined, or 'added' together.

To do so, you simply add their coefficients — the numerical parts — together. So, \[ 1x + 0.85x \] becomes \[ 1.85x \]. Notice that we treat 'x' as if it were '1x' for simplicity. This step drastically simplifies the equation and paves the way for solving it by isolating the variable 'x'. Recognizing and combining like terms efficiently is essential for streamlining the solving process and avoiding errors.

Isolating the variable, often the final hurdle in solving an algebraic equation, requires you to get the variable by itself on one side of the equation. After combining like terms, we simplified our expression to \[ 1.85x = 62.9 \]. To get 'x' by itself, we need to perform operations that will 'undo' whatever is being done to 'x'. Since 'x' is being multiplied by 1.85, we need to divide both sides of our equation by 1.85. By doing this, we 'isolate' 'x' because \[ \frac{1.85x}{1.85} \] simplifies to just 'x', and we get:
\[ x = \frac{62.9}{1.85} \].

Isolating the variable is a critical skill that helps you solve for unknowns and understand the relationship between different quantities. The process of isolating the variable can involve multiple steps, especially in more complex equations, but the principle remains the same: perform inverse operations to 'free' the variable from other terms.