91Ó°ÊÓ

To solve an algebra equation means to find the value of the variable that makes the equation true. The equation we are working with is a linear equation. This means it is of the form \(ax + b = cx + d\) where \(a\), \(b\), \(c\), and \(d\) are constants. In our particular problem, the equation is \(3x + 2 = 2x\).

The first step in solving such an equation is to get all the terms involving the variable, \(x\), on one side and the constant terms on the other side. This can be done by adding or subtracting terms from both sides. In our case, we subtracted \(2x\) from both sides to come up with \(x = 0\). This isolates the variable, making it much easier to identify a potential solution.

After finding a solution to an equation, it is important to check that the solution is correct. This verification involves substituting the found value back into the original equation.

For the equation \(3x + 2 = 2x\), we initially solved for \(x = 0\). To check this solution, substitute \(0\) into the original equation to ensure both sides are equal. Substituting yields \(3 \times 0 + 2 = 2 \times 0\). Simplifying this gives \(2 = 0\), which is not a true statement.

This contradiction indicates that \(x = 0\) is not a valid solution, hence the equation has no solutions.

Working through equations step by step can simplify the problem and help avoid mistakes. Let's break down the steps we took to solve \(3x + 2 = 2x\):

• **Identify terms:** The equation has variables \(3x\) and \(2x\), and a constant \(2\).
• **Reorganize:** Subtract \(2x\) from each side to balance the equation, resulting in \(3x - 2x = 2 - 2\).
• **Simplify:** This simplifies to \(x = 0\).
• **Verify:** Substitute \(x = 0\) back into the original equation to check; however, it shows the equation does not hold true.

Following these steps will help maintain clarity throughout the solution process.

Beginning algebra introduces concepts of variables and linear equations, which are foundational in mathematics. When dealing with equations like \(3x + 2 = 2x\), it's crucial to understand the properties of equality and basic operations.

In algebra, a variable represents an unknown value and equations are statements of equality between two expressions. Learning to manipulate these expressions to isolate the variable and find its value is a key skill.

Exercises such as these are designed to develop proficiency in handling equations, improving logical thinking, and building a groundwork that is essential in more advanced topics in mathematics.