91Ó°ÊÓ

When solving basic algebraic equations, one of the key steps is to isolate the variable. This means that we aim to have the variable by itself on one side of the equation, with no other numbers or variables attached to it. Consider the equation from our exercise, 3x - 42 = 0. To isolate the variable x, we perform actions that 'undo' whatever has been done to it. In this case, we need to cancel out the subtraction of 42.

We achieve this by adding 42 to both sides of the equation, leading to 3x = 42. This step ensures the variable x is one step closer to being isolated. It is crucial to do the same operation to both sides of the equation to maintain its balance, which brings us to our next concept of balancing equations.

The process of balancing equations is foundational in algebra and guarantees that our manipulations do not change the original meaning of the equation. To maintain the equality, whatever you do to one side of the equation, you must do to the other side.

In the provided exercise, after adding 42 to both sides, the equation is still balanced and produces an equivalent equation where the variable x is now on one side by itself. The next step requires us to eliminate the coefficient, which illustrates another key concept in solving algebraic equations.

The coefficient in an algebraic equation is the number that multiplies the variable. In the equation 3x = 42, the number 3 is the coefficient of x. To fully isolate the variable and solve the equation, we need to remove this coefficient.

This is done by performing the inverse operation of the multiplication the coefficient represents. Therefore, we divide both sides of the equation by the coefficient, which is 3 in this case. Dividing both sides by 3 leaves us with x = 14. By understanding how to manipulate coefficients, students can more effectively isolate variables and solve various algebraic equations.