Given that 3x - 5y = 22 and 4x - 3y = 33, find grafically the value of 2x + y.


First we draw the graphs of the two given lines by getting points:



Now since 2x + y has to be given the same values that x and y have where
the lines intersect, then the line

2x + y = b 

must pass through their point of intersection.

That line is equivalent to 

y = -2x + b

Comparing that to the slope-intercept form

y = mx + b

we see that it has slope -2, which we consider as 

So from their point of intersection we draw a line segment of 
length 2 downward, since -2, the numerator of the slope, is 
negative. 



Then from the end of that line we draw a line segment of length 1, 
the denominator of the slope, to the right.



Next we draw a line (not line segment) from the end of that 
line segment through the point of intersection of the original 
two lines.



Now we observe that this line crosses the y-axis at 19.

Therefore the y-intercept of the black line is 19, and since
b is the y-intercept of that line and since

b = 2x + y

the value of 2x + y must be 19.

Check:

The two original lines appear to intersect at the point 
(x,y) = (9,1), so therefore 2x + y = 2(9) + 1 = 18 + 1 = 19.

However, we found it completely graphically.

Edwin