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When solving linear equations, our goal is to find the value of the variable that makes the equation true. The process involves performing operations on both sides of the equation to maintain balance. A linear equation is an equation that, when graphed, produces a straight line. These equate forms like \(ax + b = c\), where \(a, b,\) and \(c\) are constants.
To solve any linear equation, you should follow these steps:

• Combine like terms on each side of the equation if necessary.
• Use addition or subtraction to eliminate constants from one side, focusing on isolating the variable term.
• Divide or multiply to solve for the variable.

In our example, we first subtracted 2 from both sides to begin isolating the variable term.

Variable isolation is a key step in solving equations. This involves rearranging the equation so that the variable you are solving for is by itself on one side of the equation.
This can be done by performing operations such as addition, subtraction, multiplication, or division on both sides of the equation.
For example, in our equation \(7x + 2 = 9\), we isolated the variable term \(7x\) by subtracting 2 from both sides, simplifying the equation to \(7x = 7\).
The goal is to have the variable, in this case \('x'\), appear alone with a coefficient of 1 on one side of the equation.Isolating the variable often involves reversing the operations that have been applied to it, adhering to the order of operations backward. This has a trickle effect, simplifying the problem one step at a time.

Algebraic manipulation refers to the techniques used to transform the equation to make it easier to solve. These steps can include both adding to or subtracting from both sides of the equation and multiplying or dividing both sides by the same number.
Using algebraic manipulation helps us simplify the process of solving the equations effectively. For instance, once we had \(7x = 7\), we applied algebraic manipulation by dividing each side by 7, simplifying to \(x = 1\). This helps reveal the solution.Important manipulation tips include:

• Maintain balance by performing the same operation on both sides of the equation.
• Simplify any fractions or complex expressions as early as possible.
• Check your solutions by substituting them back into the original equation.

Through these systematic actions, you simplify the equation and find the solution accurately.