Linear equations/ algebra

Question

( y=ax+b)

 

A. Why will the y intercept always be "b"?

B. Why will the slope always be "a"?

C. What will the line look like if "b" is a negative number?

D. What will the line look like if "a" is a negative number?

 

Answer 1

A.  The intercept is by definition the point where the line crosses the y-axis, i.e. where x=0.  If y=ax+b, then at x=0 y must be a(0)+b=b, so the intercept will always have coordinates (0,b).

 

B.  Suppose you have two points (x1,y1) and (x2,y2) that satisfy y=ax+b.  The slope is:

 

m = (y2-y1) / (x2-x1) = [(ax2 + b) - (ax1 + b)] / (x2 - x1) = a(x2 - x1) / (x2 - x1) = a, true for any (x,y) that meet y=ax+b

 

C.  If b is negative, that means the line crosses the y-axis below the origin.  The value of b merely shifts the line up or down without changing its slope.

 

D.  If a is negative, then y deceases as x goes from left to right.  Similarly, if a is positive, y increases as x goes from left to right.