We are given the linear equation 3x plus 3 equals 9x minus 9. Our goal is to find the value of x that makes both sides of the equation equal. This equation has a left side, 3x plus 3, and a right side, 9x minus 9.
To isolate the variable terms, we subtract 3x from both sides of the equation. On the left side, 3x minus 3x cancels out, leaving us with just 3. On the right side, 9x minus 3x gives us 6x, so we have 6x minus 9. This maintains the balance of the equation.
Now we need to isolate the constant terms. We do this by adding 9 to both sides of the equation. On the left side, 3 plus 9 equals 12. On the right side, negative 9 plus 9 cancels out, leaving us with just 6x. The equation is now 12 equals 6x.
To solve for x, we divide both sides of the equation by the coefficient of x, which is 6. On the left side, 12 divided by 6 equals 2. On the right side, 6x divided by 6 simplifies to just x. This gives us 2 equals x, or equivalently, x equals 2.
To verify our solution, we substitute x equals 2 back into the original equation. For the left side, we have 3 times 2 plus 3, which is 6 plus 3, equaling 9. For the right side, we have 9 times 2 minus 9, which is 18 minus 9, also equaling 9. Since both sides equal 9, our solution x equals 2 is correct.
Let's summarize the complete solution process. First, we started with the original equation. Second, we subtracted 3x from both sides to isolate variable terms. Third, we added 9 to both sides to isolate constant terms. Fourth, we divided by 6 to solve for x, giving us x equals 2. Finally, we verified our solution by substituting back into the original equation. The key principles used were maintaining equation balance, combining like terms, and applying inverse operations.