Theslopeof a line is the ratio of the vertical distance to the horizontal distance of a slope triangle formed by two points on a line. The vertical part of the triangle is called y(Δy) (read “change iny”), while the horizontal part of the triangle is called x(Δx) (read “change inx”). It indicates both how steep the line is and its direction, upward or downward, left to right.
Note that lines pointing upward from left to right have positive slope, while lines pointing downward from left to right have negative slope. A horizontal line has zero slope, while a vertical line has undefined slope. To calculate the slope of a line, pick two points on the line, draw a slope triangle (as shown in the example at Right), determine Δyand Δx, and then write the slope ratio. You can verify that your slope correctly resulted in a negative or positive value based on its direction. In the example above, Δy= 2 and Δx = 5, sothe slope is 2/5 .
Use DESMOS to help you find the slope of the line represented in the following table. 2-31 HW eTool.
You can find the equation of a line when you know the slope and one point on the line. To do so, rewrite y = mx + b with the known slope and substitute the coordinates of the known point forxandy. Then solve forband write the new equation.
For example, find the equation of the line with a slope of –4 that passes through the point (5, 30). Rewritey = mx + b as y=–4x + b. Substituting (5, 30) into the equation results in 30 = –4(5) +b. Solve the equation to findb= 50. Since you now know the slope andy‑intercept of the line, you can write the equation of the line asy= –4x+ 50.