91Ó°ÊÓ

Solving linear equations is a fundamental skill in algebra. A linear equation is one in which the variable(s) are raised to the power of one. The goal is to find the value of the variable that makes the equation true. For example, in the equation \(6x - 5 = 7 + 2x\), we need to determine the value of \(x\) that satisfies this equation. The process involves a series of logical steps aimed at isolating the variable on one side of the equation.

• First, you simplify both sides if necessary.
• Then, combine like terms.
• Finally, follow the steps to isolate the variable.

Let's break down these steps in detail to ensure understanding.

Isolating the variable means rearranging the equation so that the variable is on one side, and everything else is on the other. This makes it easier to see what the variable equals. In our example equation \(6x - 5 = 7 + 2x\), we want to move all the \(x\) terms to one side. We can achieve this by subtracting \(2x\) from both sides:
\[ 6x - 5 - 2x = 7 + 2x - 2x \]
This simplifies to \[ 4x - 5 = 7 \].
Now, to move the constant term (-5) to the other side, we add 5 to both sides:
\[ 4x - 5 + 5 = 7 + 5 \]
This simplifies further to \[ 4x = 12 \].
Finally, we solve for \(x\) by dividing both sides by 4: \[ \frac{4x}{4} = \frac{12}{4} \]
This gives us \[ x = 3 \. \]
By isolating the variable step-by-step, we make the solution process clear and manageable.

After solving the equation, it's vital to check that the solution is correct. Substituting \(x = 3\) back into the original equation verifies our results. The original equation is \[ 6x - 5 = 7 + 2x \].
Substitute \(x\) with 3:
\[ 6(3) - 5 = 7 + 2(3) \]
Simplify: \[ 18 - 5 = 7 + 6 \]
This results in: \[ 13 = 13 \]
Because both sides of the equation are equal, the solution \(x = 3\) is verified as correct. Checking solutions helps ensure that no mistakes were made during the solving process. It gives confidence that the answer is correct and that all steps were executed properly.