91Ó°ÊÓ

Algebraic equations are fundamental in mathematics where unknown values, represented by variables, are linked through various operations like addition, subtraction, multiplication, and division. In our given exercise, an algebraic equation is used to represent a complicated relationship between a number and its multiples. Here, we are dealing with an equation that compares the number of times a variable (in this case, "three times a number" minus 1) to twice that same number. This comparison helps us set up the equation: \(3x - 1 = 2x\). The goal is to find the value of the unknown variable, \(x\), which satisfies the equation. Understanding algebraic equations involve learning how to manipulate these variables and constants to isolate \(x\) and solve the problem at hand.

Mathematical expressions are combinations of numbers, operators, and sometimes variables. They represent values without an equality sign, unlike equations. In our exercise, we encounter two main expressions: \(3x - 1\) and \(2x\). Each of these expressions involves a variable \(x\), representing the unknown number. While the expression \(3x - 1\) conveys the idea of taking three times the number and subtracting one, \(2x\) means simply doubling the number. Learning how to read and translate word problems into mathematical expressions is essential. This skill allows you to recognize and work with the arithmetic relationships described in a problem, and it is a critical step towards solving algebraic equations.

Verifying solutions is the final and crucial step in solving any algebraic equation. It assures us that the computed result is correct. After solving for \(x\) using an algebraic equation, it's necessary to substitute this value back into the original expressions to ensure they hold true.For example, substituting \(x = 1\) into our exercise checks like this:- Three times the number minus one is \(3(1) - 1 = 2\).- Twice the number is \(2(1) = 2\).Both calculations confirm that the expressions are equal, verifying that our solution \(x = 1\) is indeed correct. This step helps to catch any potential mistakes made during the process of solving the equation and ensures the integrity of your solution.