This a lot to explain while still being simple. I will try to be as thorough as I can, while keeping the answer simple. It may help if you gave a specific problem as a reference to what you need help with.
Linear equations are equations with two variables (usually x and y) in the first power (no exponants or square root signs).
The term linear equation comes from the fact that the graph of the equation is a line. There are an infinite number of ordered pairs that are correct for the equation, but there is only one x for each y and only one y for each x (except for vertical and horizontal lines).
There are several ways to write a linear equation, but the simplest is probably slope intercept. The slope intercept form of linear equations is given by the general formula y = mx + b. In this formula, y and x remain as the variables, but m and b are replaced with numbers.
When dealing with the slope intercept form of a line, m is the slope. To find the slope on a graph, choose two known points. Starting at the point on the left, count how much you go up or down to reach the second point. This becomes the numerator of the slope (the sign will be negative if you have to go down). Next, count how many places you move to the right. The becomes the denominator of the fraction. Now, reduce your fraction (if possible) and you have the slope.
Example: I have the points (0, 4) and (3, 6)
Going from the first point to the second point, I have to count up 2. The numerator in my fraction is 2.
Going from the first point to the second point, I have to count right 3. The denominator of my fraction is 3.
The slope (m) is 2/3.
Back to slope-intercept form...
The b in slope-intercept form is called the y-intercept. It is the value of y when x = 0. To find this value on a graph, just look at how high the line is when it crosses the y-axis (bold vertical line where x=0).
Example (cont'): I am still using the points (0, 4) and (3, 6)
Since the y-intercept is the value of y when x = 0, the y-intercept (b) of this line is 4.
Since I have the slope and y-intercept of the line, I can now write the answer in slope intercept form by substituting in my values for m and b.
m = 2/3, b = 4
y = 2/3 *x + 4
I am going to stop my answer at this point and give you a chance to look it over and see if it helped. I hope that I have at least cleared up a little of your confusion.