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Solving equations is a fundamental skill in algebra. The main goal is to find the value of the variable that makes the equation true. In this exercise, the variable is \( x \). Solving an equation often involves a sequence of steps to simplify and manipulate the expression until it clearly shows the solution.The process can include:

• Removing fractions or decimals for simpler calculations
• Eliminating numbers from one side to isolate the variable
• Using operations such as addition, subtraction, multiplication, or division

In our example, the equation \( \frac{2 x}{9} + 1 = \frac{7}{9} \) relies on simplification and isolation techniques to find \( x \). By following these steps, the solution becomes clear and straightforward.

In algebra, expressions consist of numbers, variables, and operations. Algebraic expressions form the basis of equations, like the one we have here: \( \frac{2x}{9} + 1 = \frac{7}{9} \).An algebraic expression can be as simple as a single variable or as complex as a combination of several terms interconnected by addition, subtraction, multiplication, or division.

• The expression \( \frac{2x}{9} \) signifies \( x \) multiplied by 2, then divided by 9.
• The term \( +1 \) merely adds to this fraction to form a complete side of the equation.
• Algebraic expressions are not equations until they contain an equality sign.

Understanding how to manipulate algebraic expressions correctly, allows us to effectively solve for the unknown in the equations, just like our given problem.

The isolation technique is crucial for solving equations. It involves rearranging the equation so that the variable you're solving for is on one side of the equation, and everything else is on the other side.Let's see how this works in our sample equation:- Start by removing constant numbers that aren't attached to the variable. In our case, subtract 1 from both sides to deal with \( \frac{2x}{9} = \frac{7}{9} - 1 \).- Simplifying the equation gives \( \frac{2x}{9} = \frac{-2}{9} \). Now we're closer to isolating \( x \).By continuing:- Eliminate the fraction by multiplying both sides by 9, resulting in \( 2x = -2 \).- Finally, divide by 2 to completely isolate \( x \), leaving us with \( x = -1 \).This step-by-step approach makes it much easier to see the path to the solution, helping to minimize errors along the way.