Infinidade de soluções

Infinity and Beyond!

You may have heard of infinity and beyond from a favorite toy adventurer...

...but we're not really talking about flying here. Instead, we're talking about systems of equations. Most systems of equations in Algebra 1 (where we are now) will have just one solution—just one set of variable values where all the equations are true. You'll also have systems with no solutions. But some systems have infinite solutions.

How can systems have infinite solutions? If you solve a system with graphing, you might notice that the solution is where the lines cross. Let's look at a system of equations that has infinite solutions:

$$\begin{gather} 22x+11y=44\\ 14x+7y=28 \end{gather}$$

Try simplifying the top equation. What do you notice?

$$\begin{gather} 2x+y=4 \\ 2x+y=4 \end{gather}$$

These are same equation! They'll both be exactly the same line on a graph. In other words, they'll overlap completely. This is easier to tell because both equations are in standard form.

You'll notice that if you convert from standard form to slope-intercept form, you'll see that they both have the same slope!

The Graph

Viola!

Every point on the line will satisfy both equations. The line extends infinitely, so there are an infinite number of solutions.

Systems with Infinitely Many Solutions

There is always a possibility that a system of equations has infinite solutions.

While this sounds like a really BIG idea, there's a pretty simple way to tell if your system of equations has infinite solutions.

If the equations in your system are the same, then there are infinite solutions.

This is why it's important to simplify your equations as much as possible before you really dig into a problem.

Let's take a look at an example:

$$\begin{gather} 2x + 5y = 16\\ 10y = 32 - 4x \end{gather}$$

Initially, those equations look pretty different. Rearranging the second equation so that the $x$ and $y$ terms are on the same side gives us $4x+10y=32$. This makes it more clear that we can reduce it further. We can factor out a common factor of $2$, which gives us equal equations.

$$\begin{align} 2x + 5y = 16\\\ 2x + 5y = 16 \end{align}$$

Our system of equations can be graphed, although instead of showing two separate lines, there will only be one.

We can think of this graph as one line placed directly on top of another. They overlap at all of their points.

The point where the lines in a system of equations intersect is the solution.

When your system includes two identical equations, the lines overlap at ALL points. Imagine the blue line in the graph extending forever and ever in both directions. Both lines would still overlap. Because the equations overlap at all points, all points on the line are possible solutions. This is what we call infinite solutions.

Note, though, that having infinite solutions isn't the same as ANY solution. There are still lots of numbers that won't work! For example, if we have the solution $(0,0)$, this will cause both equations to be $0 = 16$, which is not true! There are infinitely many solutions, but all solutions must be on the line.

(Video) Systems of Equations with Infinitely Many Solutions

by The Math Sorcerer

This video above by The Math Sorcerer shows linear systems with infinitely many solutions.

$\large{\text{Summary}}$

Suppose we have the system: $$\begin{align} x+y&=4\\ 3x+3y&=12 \end{align}$$

Let's multiply all terms in the first equation by $-3$ so that we can add and cancel variables. $$\begin{align} x+y&=4\\ -3(x+y)&=-3(4)\\ -3x-3y&=-12\\ 3x+3y&=12 \end{align}$$

Now our new system is:

$$\begin{align} 3x+3y&=12\\ 3x+3y&=12 \end{align}$$

These are the same equation!! If we graphed them, they would be the same line.

When we graph systems of equations, the intersection of the lines is the solution. If a system has infinitely many solutions, then the lines overlap at every point. In other words, they're the same exact line! This means that any point on the line is a solution to the system. Thus, the system of equations above has infinitely many solutions.