Algebra Know-It-ALL

642 Worked-Out Solutions to Exercises: Chapters 11 to 19

The slope of this graph is −2, the same as the slope of the graph of the other equation.

But the y-intercept is 4, and that’s different. If we plot the graphs, we get two parallel

lines. On the Cartesian plane, two lines have the same slope but different y-intercepts if

and only if they’re parallel. Now remember from plane geometry: parallel lines do not

intersect. That means they have no point in common. When two parallel lines appear on

the Cartesian plane, no ordered pair (x,y) can give us a point that falls on both lines, so

no ordered pair (x,y) can satisfy both equations.

7. Let’s put the two equations from Probs. 5 and 6 into the format we used to solve the
   challenge in the section “Double Elimination.” Here again are those general equations:

ax+by=c

and

dx+ey=f

where x and y are the variables, and a through f are constants. Here are the two equations

from Probs. 5 and 6 that we could not solve as a linear system:

2 x+y= 3

and

6 x+ 3 y= 12

We have a= 2, b= 1, d= 6, and e= 3. Therefore,

ae= 2 × 3

= 6

and

bd= 1 × 6

= 6

Now remember that in the general derivation, we were not allowed to let ae=bd, because

that would cause us to divide by 0 in the course of trying to solve the system. You’ve

already seen some of the bad things that can happen when we divide by 0 directly or

indirectly, knowingly or unknowingly. Let this serve as another example!

8. Let’s call the numbers x and y, as we did in Prob. 1. The substitution process is similar
   to the morph-and-mix process, at least for this system. Here are the equations again:

x+y= 44

and

x−y= 10