91Ó°ÊÓ

The slope of a line describes its steepness and is represented by the letter \(m\).
The slope is calculated as the 'rise over run', or the change in the y-coordinate divided by the change in the x-coordinate between two points.
For example, in the line equation \(y = \frac{1}{2}x + 2\), the slope is \(\frac{1}{2}\), meaning for every 2 units you move to the right on the x-axis, you will move up 1 unit on the y-axis.
Slopes can be positive, negative, or even zero. A positive slope means the line goes up as you move to the right. A negative slope means the line goes down. A zero slope is a horizontal line. For instance, a slope of 2 means you will rise 2 units for each unit you move to the right.

The y-intercept is where a line crosses the y-axis. This point is crucial to graphing linear equations.
It is represented by the letter \(c\) in the equation \(y = mx + c\).
In the given problem, all lines pass through the y-intercept \(c = 2\), where they cross the y-axis at the point (0,2).
To find the y-intercept from the equation, set \(x = 0\). If your equation is \(y = \frac{5}{6}x + 2\), substitute 0 for \(x\), which makes \(y = 2\).
Thus, the graph touches the y-axis at 2. Always start your graph at the y-intercept.
This strategy simplifies plotting the entire line.

The point-slope form of a linear equation helps easily graph a line when you know the slope and a point on the line.
It is generally written as \ (y - y_1) = m(x - x_1) \. Here, \(m\) is the slope and \( (x_1,y_1) \) is the given point.
For the point (0,2) and slope \ (\frac{1}{2}) \, the equation becomes \ y - 2 =\frac{1}{2} (x - 0) \.
Solving for \ y \, you get \ y =\frac{1}{2}x + 2 \.
You can apply this formula to any slope and point. This form makes it easier when lines have various slopes but share the same y-intercept.
Hence, each of the given equations starts with y = and ends with 2 as the y-intercept is fixed at that value.

Linear equations graph as straight lines and have the form \ y = mx + c \. Here, \(m\) is the slope and \(c\) is the y-intercept.
They represent the relationship between two variables resulting in a straight line when plotted.
The general solution steps: find the y-intercept, use the slope to determine the next points, and draw the line.
In these exercises, you plot each equation by starting at the y-intercept (0,2), and using the slopes \ (\frac{1}{2}), (2), (\frac{5}{6}) \ to find subsequent points.
Each line must pass through the y-intercept and extend left/right according to its slope.
Practicing with different slopes can help understand how lines behave in a graph and aid in solving linear equation problems.