91Ó°ÊÓ

To start, it's essential to know what point-slope form is. The point-slope form of a linear equation is represented as: \( y - y_1 = m(x - x_1) \). Here, 'm' is the slope of the line, and \((x_1, y_1)\) is a specific point through which the line passes.

Using point-slope form is especially useful when you know the slope of the line and one point on the line. For example, if you're given the equation \( y + 2 = -3(x - 4) \), you can identify the slope (-3) and the point (4, -2). This sets the foundation for converting it to slope-intercept form.

It's an intermediate step that makes it easier to later isolate \( y \) and express the line equation in a more familiar and useful form.

Now, let's discuss the distributive property. This mathematical property is used to multiply a single term across terms inside a parenthesis. The general form is: \( a(b + c) = ab + ac \).

In our example equation, \( y + 2 = -3(x - 4) \), we apply the distributive property to simplify it:
\( y + 2 = -3 \times x + (-3) \times (-4) \).
Let's break it down. We multiply -3 by x, which gives us -3x, and then -3 by -4, which results in +12.

Therefore, the equation becomes: \( y + 2 = -3x + 12 \).

The distributive property is handy for eliminating parenthesis, allowing us to simplify and rearrange the equation more easily.

Finally, it's time to isolate the variable. Here, our goal is to isolate \( y \) to write the equation in slope-intercept form (\( y = mx + b \)).

Starting with the simplified equation \( y + 2 = -3x + 12 \), we need to move the constant term (2) from the left side to the right. To do this, subtract 2 from both sides of the equation:
\( y + 2 - 2 = -3x + 12 - 2 \)

Simplifying the subtraction, we get: \( y = -3x + 10 \).

By isolating \( y \), we get the equation in the slope-intercept form (\( y = mx + b \)), which is: \( y = -3x + 10 \).

Isolating variables is crucial in algebra because it helps to express equations in their simplest forms, making them easier to work with and understand.