Choose any two points on a line to determine the slope of the line. If the points are (x1, y1) and (x2, y2), the slope will be (y1 - y2) / (x1 - x2). You can think of this as drawing a little triangle beneath the line. The height of the triangle is y1 - y2 and the base of the triangle is x1 - x2. The hypotenuse is a the line segment between the two points. If the line were turned so that it became steeper, the height of the triangle would become larger with respect to the base -- so the slope increases.
Find a line parallel to a given line and through a given point. If the formula for the given line is y = ax + b, in which "a" is the slope and "b" is the y-intercept, the slope of both lines is the same. If the new line goes through point (x1, y1), you can put x1 and y1 in the formula to get y1 = ax1 + b and solve for b to get the new formula. For example, the line through the point (2, 5) and parallel to the line y = 3x + 1 would be y = 3x + b. To find the b, solve 5 = 3 * 2 + b to get b = -1, so the new line is y = 3x -1.
Draw the line through points (x1, y1) and (x2, y2) by finding the "a" and "b" in y = ax + b. The slope is a = (y1 - y2) / (x1 - x2). Find the "b" by substituting either point in the equation and solving for b. For example, The line through points (0, 3) and (1, 5) has slope (3 - 5) / (0 -1) = -2/-1 = 2. Substituting in y = 2x + b the point (0, 3), you get 3 = 0 + b, so the desired line is y = 2x + 3.