GMAT Official Guide Quantitative Review 2019_ Book

3.3 Geometry

y

(-^2 ,2)

-3 - 2 -1

(2,0)

． ．

1 2 3 X

• 1

In the equation y = mx + b of a line, the coefficient m is the slope of the line and the constant term b

is they-intercept of the line. For any two points on the line, the slope is defined to be the ratio of the

difference in they-coordinates to the difference in the x-coordinates. Using (-2, 2) and (2, 0) above, the

slope is

The difference in the y-coordinates ＝＝O - 2 - (^2) ＝－－ 1
The difference in the x-coordinates 2- (-2) 4 2
They-intercept is they-coordinate of the point at which the line intersects the y-axis. For the line
above, they-intercept is 1, and this is the resulting value of y when xis set equal to O in the equation
y = - - 2 1 x + 1. The x-intercept is the x-coordinate of the point at which the line intersects the x-axis. The
x-intercept can be found by setting y = 0 and solving for x. For the line y = --^1 x + 1, this gives
2

• -^1 x+l=O
  
  
  • -^1 x=- 1
  
  
  
  

x=2.

Thus, the x-intercept is 2.

Given any two points (x可 1 )and (x初 2 )with x1 "#- x2, the equation of the line passing through these

points can be found by applying the definition of slope. Since the slope is m = Y2- Y1 , then using a

X2 -Xl

point known to be on the line, say (x议1),any point (x,y) on the line must satisfy y-Y1= m, or

x-x

y -Yi = m(x -xi). (Using (x动2)as the known point would yield an equivalent equation.) For example,
consider the points (-2,4) and (3,-3) on the line below.