Algebra Know-It-ALL

276 Two-by-Two Linear Graphs

When taken together as a two-by-two linear system, these equations have no common

solution. That means their graphs do not intersect, so we can’t use their point of intersec-

tion as a graph-plotting aid. But there’s another way. Both lines have slopes of −2. If we

move n units to the right along either line (where n is any number), we must move 2n units

downward to stay on the line. Let’s move to the right by 4 units along each line. That means

we must go downward by 8 units. Starting at (0, 3), we’ll end up at (4, −5). Starting at (0, 4),

we’ll end up at (4, −4).

Connect the points

Figure 17-7 shows the four points we’ve found, and the two lines connecting them. Note that

the lines are parallel, so they have no intersection point. These two lines, taken together, form

the graph of the two-by-two linear system represented by the inconsistent equations

2 x+y= 3

and

6 x+ 3 y= 12

x

y

(0,3)

(0,4)

(4,–5)

(4,–4)

2 x+y= 3

y= –2x+ 3

6 x+ 3y= 12

y= –2x+ 4

Each axis

increment

is 1 unit

Figure 17-7 Graphs of 2x+y= 3 and 6x+ 3 y= 12

as a two-by-two linear system where the

independent variable is x and the dependent

variable is y. The lines are parallel and

distinct. On both axes, each increment

represents 1 unit. The SI forms of the

equations are shown below the originals.