91Ó°ÊÓ

In mathematics, especially when dealing with linear equations, one of the key goals is to isolate the variable. This means getting the variable by itself on one side of the equation. For example, in the equation given in the problem, \(-50 = 7n - 1\), we wanted to isolate the variable \(n\). To do this, we perform various operations that alter the equation but keep it balanced. Starting by adding, subtracting, multiplying, or dividing both sides of the equation as needed helps in getting the variable alone.

Isolating variables simplifies solving equations because it helps to clearly identify what the variable equals. It's like peeling away layers to reveal the core value you're solving for. This process involves using properties like the addition property of equality and the division property of equality to manipulate the equation correctly.

The addition property of equality states that if you add the same amount to both sides of an equation, the equality is still true. This is useful because it helps balance the equation while isolating the variables. For example, in the original problem \(-50 = 7n - 1\), we added 1 to both sides:

• \(-50 + 1 = 7n - 1 + 1\)

This simplifies to <\(-49 = 7n\). By doing this, we effectively moved closer to isolating \(n\).

It's essential to apply this property accurately because any change made to one side must be applied to the other side to maintain the balance of the equation. This ensures we do not change the equation's meaning but simply refine it to make solving easier.

The division property of equality states that you can divide both sides of an equation by the same nonzero number, and the two sides will remain equal. This property is incredibly handy for isolating variables that have been multiplied by a coefficient. In the example from the exercise, after adding 1 to both sides, we had \(-49 = 7n\). To isolate \(n\), we used division:

• \(\frac{-49}{7} = \frac{7n}{7}\)

Here, dividing both sides of the equation by 7 simplified it to \(n = -7\).

Division by the coefficient achieves the goal of isolating the variable, making this property a vital tool in solving linear equations. It's essential to remember that division must be done on both sides to keep the equation balanced, just like with addition, subtraction, or multiplication.