91Ó°ÊÓ

The slope-intercept form of a line's equation is written as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, where the line crosses the y-axis.
This form is useful because it easily shows the slope and intercept at a glance.

To find the slope-intercept form, you need the slope and at least one point on the line.
You often start from another form, like point-slope form or standard form, and then solve for \(y\) to get it into slope-intercept form.
In the above exercise, we start from the point-slope form and rearrange the equation to achieve the slope-intercept form, \(y = \frac{1}{2}x + 1\). This makes it easy to recognize the slope (\(\frac{1}{2}\)) and the y-intercept (1).

The point-slope form of a line's equation is given by \(y - y_1 = m(x - x_1)\).
Here, \(m\) is the slope of the line, and \((x_1, y_1)\) is a specific point on the line.
This form is particularly useful when you know a point on the line and the slope but don’t have the y-intercept.

For instance, in our solution, given the slope \(\frac{1}{2}\) and the point (2,2), we used the point-slope form: \(y - 2 = \frac{1}{2}(x - 2)\).
To convert this to the slope-intercept form, we simply solve for \(y\). Distributing \(\frac{1}{2}\) gives us \(y - 2 = \frac{1}{2}x - 1\), and adding 2 to both sides results in \(y = \frac{1}{2}x + 1\).
Point-slope form is a powerful tool because it makes deriving the equation of the line straightforward when you have a slope and a single point.

To determine the slope of a line, you can use the slope-intercept form \(y = mx + b\).
The value of \(m\) is your slope. In a given equation like \(y = -2x + 3\), the slope (\(m\)) is -2.

If you need to find the slope of a line perpendicular to it, remember that the slopes of perpendicular lines are negative reciprocals.
This means if one line's slope is \(m\), the perpendicular line's slope will be \(-\frac{1}{m}\).

Applying this, if the slope of the given line is -2, the slope of the line perpendicular to it is \(\frac{1}{2}\), because \(-2 \times \frac{1}{2} = -1\).
Identifying slopes correctly is essential in ensuring the accuracy of your line equations, especially when dealing with perpendicular lines.