No Solution, or Infinitely Many

In this lesson you will: Compare slopes and intercepts to predict how many solutions a system has; Read a false statement as no solution and a true statement as infinitely many; Say what each outcome means about the two lines and about the situation they model.

Three pictures, and only three

Two lines in a plane can do exactly three things. They can cross once, they can run parallel and never meet, or they can be the same line drawn twice.

You met this argument in Chapter 3, where an equation with the variable on both sides collapsed to a value, to something false, or to something true. It is the same trichotomy, now with two variables and two lines to look at.

What the collapse looks like in algebra

You will not always be handed slope-intercept form. Run substitution or elimination as usual; the system tells you which case you are in by what survives.

For 2x+3y=6 and 4x+6y=18, multiply the first by 2: 4x+6y=12. Set beside 4x+6y=18, this says 12=18. False, so there is no solution — the two lines are parallel.

For 3x-y=4 and -6x+2y=-8, multiply the first by -2: -6x+2y=-8. That is the second equation, character for character. The two equations are the same line, so every point on it is a solution.

What it means in a situation

These outcomes are not curiosities. If two payment plans charge the same rate and differ only in their start-up fee, no number of units makes them cost the same — the cheaper one is cheaper forever.

If two descriptions of a situation turn out to be the same equation in different clothing, you have been given one piece of information twice, and it will not be enough to pin down a unique answer.

Common mistake — Turning a collapsed equation into a value of x

A system reduces to 0=0, and the answer written down is x=0.

It is an easy slip to make. Almost every system you have solved ended with a variable equal to a number, so a zero sitting alone at the end of the work gets read as that number. The mirror-image version is reading 0=7 as x=7.

But 0=0 has no x in it. It is not a claim about one value; it is a claim that holds no matter what x is, which means every point on the line is a solution. And 0=7 is false no matter what x is, which means no point is.

How to catch it: Look at the line you finished on and ask whether a variable still appears in it. If not, you have learned something about the whole system, not about one coordinate. True statement → infinitely many solutions. False statement → none.