Algebra Know-It-ALL

If we solve this system, we’ll get two results, but they’ll both be non-real complex numbers. For extra

credit, you can solve the “revised” system and see for yourself.

Are you confused?

Does the slope of the line in Fig. 28-1 seem smaller than 2? In a way, it is! In the algebraic sense the slope

is 2, but in the geometric sense it’s only 2/5. The increments on the vertical axis are five times as large as

the ones on the horizontal axis. That distorts the slopes and contours of the graphs, expanding everything

horizontally (or compressing everything vertically) by a factor of 5. If we had used a true Cartesian plane,

the line would have the steepness we should expect for a slope of 2 when drawn. The parabola would also

look different; it would seem “five times sharper.”

Two Quadratics

In the second example in Chap. 27, we solved this system of quadratic equations in two variables:

4 x^2 + 6 x+ 2 y+ 8 = 0

and

3 x^2 +y+ 5 x− 11 = 0

466 More Two-by-Two Graphs

x

y

(–2,–3)

(3,7)

Figure 28-1 Graphs of y= 2 x+ 1 and y=x^2 +x− 5. The first

function is graphed as a solid line; the second function

is graphed as a dashed curve. Real-number solutions

appear as points where the line and the curve intersect.

On the x axis, each increment is 1 unit. On

they axis, each increment is 5 units.