Finding y-intercepts

Date: 6/12/96 at 19:2:1
From: Ourpond
Subject: Linear Equations

Dear Dr. Math:

I'm having a problem with finding y-intercepts. How do you know what 
the y-intercept is for the following equations?

2x + y = 3 or x - 4y + 8 = 0 

I'm also having trouble with this: Draw the graph of each line: 
x-intercept = 2, slope = 3/4                                     

Thanks, Tanya 

Date: 6/13/96 at 20:58:29
From: Doctor Tom
Subject: Re: Linear Equations

Hi Tanya,

The "y intercept" is the place where the graph hits the y-axis.  If
you think about what the y axis is, it represents all the points on
the plane where the x value is zero, right?

Putting this in another way, to find a point given the x and y 
coordinates, you move left or right by the x amount, and then up or 
down by the y amount.  To stay on the y axis, you make no motion in 
the x direction; in other words, the x-coordinate is zero.

So to find the y-intercept of "2x+y=3", it occurs when x = 0, or when
2*0+y=3, or when y=3.  For "x-4y+8=0", set x=0, making -4y+8=0, or
4y=8, or y=2.  That's all there is to it.

For your second problem, to draw a line with x-intercept = 2 and slope
of 3/4, you know one point already - on the x-axis at the point 2.

A slope of 3/4 means that for every 4 units you move in the 
x direction, the line goes up 3 units in the y-direction.  
So starting from the one point you know, (at (2,0)), move to 
the right 4 units and up 3 units to the point (6,3), and that's 
another point on the line.  Just draw the straight line through those 
points.

To convince yourself that the slope concept makes sense, go over
another 4 and up another 3 to (10,6), then repeat it to (14,9), and
notice that all those points lie on the line.

I think it's very important to have this sort of a geometrical, or
visual idea of a slope.  The numerator represents the motion in the
y-direction, and the denominator, the motion in the x-direction.  If
the slope is 5, that's the same as 5/1 - numerator of 5 and 
denominator of 1.  And negative slopes simply indicate that instead of 
sloping up, the line slopes down as you move to the right.

I hope this helps.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/