91Ó°ÊÓ

When solving linear equations, such as \(4-2x=22\), the goal is to find the value of the variable that makes the equation true. To do this, we 'isolate' the variable, which means we manipulate the equation to get the variable by itself on one side.

The solution begins with adding \(2x\) to both sides to cancel out the negative term and move the variable to the right side, resulting in \(4 = 22 + 2x\). It's like balancing scales: what you do to one side, you must do to the other to maintain equality. The key to isolating the variable is to reverse the operations that are being applied to the variable. This process requires understanding the inverse operations: addition and subtraction are opposites, as are multiplication and division.

To simplify the equation, we perform operations that will bring us closer to isolating the variable. After moving the '2x' to the right side, we subtract 22 from both sides to remove the constant term from the right side, leaving us with \(4 - 22 = 2x\).

By simplifying this, we combine like terms which results in \(-18 = 2x\). It's crucial to follow the order of operations and to combine only like terms to simplify the equation correctly. Simplifying sets the stage for us to finish solving the equation by clearing the coefficient of the variable, which in this case is 2.

The final steps in solving equations involve performing operations to isolate the variable. After simplifying our example to \(-18 = 2x\), we need to deal with the coefficient of \(x\), which is the number by which the variable is multiplied.

We divide both sides by 2, the inverse operation of multiplication, to isolate \(x\). This gives us: \[x = \frac{-18}{2}\] Simplifying the division, we find \(x = -9\). Division is a critical operation in solving equations because it allows us to get the variable alone and solve for its value. It's important to apply the same operation to both sides of the equation to maintain its balance and to reach the correct solution.