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Special Cases

Module by: Kenny M. Felder. E-mail the author

Summary: This module discusses concepts related to simultaneous equations.

Consider the two equations:

2x + 3y = 8 2x + 3y = 8 size 12{2x+3y=8} {}
(1)
4x + 6y = 3 4x + 6y = 3 size 12{4x+6y=3} {}
(2)

Suppose we attempt to solve these two equations by elimination. So, we double the first equation and subtract, and the result is:

4 x + 6 y = 16 4 x + 6 y = 3 0 = 13 4 x + 6 y = 16 4 x + 6 y = 3 0 = 13
(3)

Hey, what happened? 0 does not equal 13, no matter what x x is. Mathematically, we see that these two equations have no simultaneous solution. You asked the question “When will both of these equations be true?” And the math answered, “Hey, buddy, not until 0 equals 13.”

No solution.

Now, consider these equations:

2x + 3y = 8 4x + 6y = 16 2x + 3y = 8 4x + 6y = 16 alignl { stack { size 12{2x+3y=8} {} # size 12{4x+6y="16"} {} } } {}
(4)

Once again, we attempt elimination, but the result is different:

2x + 3y = 8 4x + 6y = 16 0 = 0 2x + 3y = 8 4x + 6y = 16 0 = 0 size 12{alignl { stack { { {2x+3y=8 {} # } over {4x+6y="16" {} # } } } { {" "0=0} } {}
(5)

What happened that time? 0=00=0 size 12{0=0} {} no matter what xx size 12{x} {} is. Instead of an equation that is always false, we have an equation that is always true. Does that mean these equations work for any xx size 12{x} {} and yy size 12{y} {}? Clearly not: for instance, (1,1)(1,1) size 12{ \( 1,1 \) } {} does not make either equation true. What this means is that the two equations are the same: any pair that solves one will also solve the other. There is an infinite number of solutions.

Infinite number of solutions.

All of this is much easier to understand graphically! Remember that one way to solve simultaneous equations is by graphing them and looking for the intersection. In the first case, we see that original equations represented two parallel lines. There is no point of intersection, so there is no simultaneous equation.

Figure 1
Graph showing two functions as parallel lines as they do not cross. For every x, there is no value that will give the same result for both functions.

In the second case, we see that the original equations represented the same line, in two different forms. Any point on the line is a solution to both equations.

Figure 2
Graph illustrating two functions in two different forms but representing the same line. For every value x, the function will yield the same result when written in slope-intersect form.

General Rule:

If you solve an equation and get a mathematical impossibility such as 0 = 13 0 = 13 , there is no solution. If you get a mathematical tautology such as 0 = 0 0 = 0 , there is an infinite number of solutions.

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