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
