Find and compare the slope between each pair of points. What does it mean if the slopes are all the same?
Linear? Yes, see solution. Equation: y=-3x+7
Practice makes perfect
One way to determine if points are collinear is to calculate and compare the slope of the line between points in the data set. We can do this by using the Slope Formula. Note that it is enough to check each point with just one other point in the data set.
Rate of Change Between...
y_2-y_1/x_2-x_1
m
( -3, 16)&( -1, 10)
10- 16/-1-( -3)
-3
( -1, 10)&( 1, 4)
4- 10/1-( -1)
-3
( 1, 4)&( 3, -2)
-2- 4/3- 1
-3
( 3, -2)&( 5, -8)
-8-( -2)/5- 3
-3
We see that the slope between each pair of points is m= -3. Since the rate of change is constant, we know that the points are collinear. To write an equation for this line, we will use the point-slope form.
y- y_1=m(x- x_1)
In this form, m is the slope of the line and the point ( x_1, y_1) lies on the line. Let's substitute the slope we calculated above and one of the given points, ( -3, 16), into this equation.
y- 16=-3(x-( -3))
Note that any point that lies on the same line as the given points would create a valid equation for this line. To write a unique linear equation that represents y as a function of x, we will rewrite this point-slope form equation in slope-intercept form.
y=mx+ b
In this form, m represents the slope of the line and b represents the y-intercept.
