91Ó°ÊÓ

When solving linear equations, the addition property of equality is a fundamental concept to understand. This property states that you can add the same number to both sides of an equation without changing the equation's balance. In other words, if you have an equation like \(a = b\), you can add any number \(c\) to both sides and the equality will still hold true: \(a + c = b + c\).

For example, in the equation given in the exercise, \(7x - 8 = 34\), you can add 8 to both sides to help isolate the term containing the variable. This gives us \(7x - 8 + 8 = 34 + 8\), which simplifies to \(7x = 42\).

Understanding this property is crucial because it allows you to systematically simplify and solve equations by systematically removing terms from one side.

The isolation of a variable is a key step in solving equations. This involves manipulating the equation so that the variable (in this case, \(x\)) is alone on one side of the equation. Isolating the variable allows you to find its value more easily.

In our example, after using the addition property of equality to simplify the equation to \(7x = 42\), the next step is to isolate \(x\). To do this, we need to get \(x\) by itself on one side of the equation by using inverse operations. Initially, \(x\) is multiplied by 7. To isolate \(x\), we need to perform the inverse operation, which leads us to the next property: division.

The division property of equality states that you can divide both sides of an equation by the same nonzero number and the equality will still hold. This property helps in solving equations where the variable has a coefficient (a number multiplied by the variable).

Continuing with our example, we reached \(7x = 42\). To isolate \(x\), we divide both sides by 7, the coefficient of \(x\): \[ \frac{7x}{7} = \frac{42}{7} \].

This simplifies to \(x = 6\).

By dividing both sides by the coefficient of \(x\), we effectively isolate \(x\) and solve the equation. This simplifies the problem to finding the value of \(x\), which in this case is 6. Understanding the division property is essential for solving equations efficiently.