Exercise 37 Page 170

We have been given the points (0,b) and (1,b+m) and asked to show that the equation of this line is y=mx+b. The slope-intercept form of a line can be thought of as: y=Slope* x+the y-intercept. The y-intercept of any line is the point at which the line crosses the y-axis. This is always when x=0. Therefore, having been given that the point (0,b) lies on the line, we know that when x=0, y=b and our y-intercept is b. Our equation then becomes: y=Slope* x+ b. We can use the Slope Formula and the given points to find the slope.

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 0,b) & ( 1,b+m)

m=b+m- b/1- 0

AddSubTerms

Add and subtract terms

m=m/1

DivByOne

a/1=a

m=m

The entire equation is then: y= mx+ b.

How can we be sure about (-1,b-m)?

Let's substitute the point (-1,b-m) into the equation so that we can test whether the statement holds true.

y=mx+b

SubstituteII

x= -1, y= b-m

b-m? =m( -1)+b

MultNegPos

(- a)b = - ab

b-m? =- m+b

CommutativePropAdd

Commutative Property of Addition

b-m=b-m

We have reached an identity statement, b-m will always equal b-m. Therefore, we know that the point lies on the line no matter the values for b and m.