3.1 Simultaneous equations by graphing

Consider the equation $y=2\sqrt{x}$ . How many $(x,y)$ pairs are there that satisfy this equation? Answer: $(\mathrm{0,0})$ , $(\mathrm{1,2})$ , $(\mathrm{4,4})$ , and $(\mathrm{9,6})$ are all solutions; and there is an infinite number of other solutions . (And don’t forget non-integer solutions, such as $(,1)$ !)

Now, consider the equation $y=x+\frac{1}{2}$ . How many pairs satisfy this equation? Once again, an infinite number. Most equations that relate two variables have an infinite number of solutions.

To consider these two equations “simultaneously” is to ask the question: what $(x,y)$ pairs make both equations true ? To express the same question in terms of functions: what values can you hand the functions $2\sqrt{x}$ and $x+\frac{1}{2}$ that will make these two functions produce the same answer ?

At first glance, it is not obvious how to approach such a question-- it is not even obvious how many answers there will be.

One way to answer such a question is by graphing. Remember, the graph of $y=2\sqrt{x}$ is the set of all points that satisfy that relationship; and the graph of $y=x+\frac{1}{2}$ is the set of all points that satisfy that relationship. So the intersection(s) of these two graphs is the set of all points that satisfy both relationships .

How can we graph these two? The second one is easy: it is a line, already in $y=mx+b$ format. The $y$ -intercept is $\frac{1}{2}$ and the slope is 1. We can graph the first equation by plotting points; or, if you happen to know what the graph of $y=\sqrt{x}$ looks like, you can stretch the graph vertically to get $y=2\sqrt{x}$ , since all the $y$ -values will double. Either way, you wind up with something like this:

We can see that there are two points of intersection. One occurs when $x$ is barely greater than 0 (say, $x=0\text{.}1$ ), and the other occurs at approximately $x=3$ . There will be no more points of intersection after this, because the line will rise faster than the curve.

Graphing has three distinct advantages as a method for solving simultaneous equations.

• It works on any type of equations.
• It tells you how many solutions there are, as well as what the solutions are.
• It can help give you an intuitive feel for why the solutions came out the way they did.

However, graphing also has two dis advantages.

• It is time-consuming.
• It often yields solutions that are approximate , not exact—because you find the solutions by simply “eyeballing” the graph to see where the two curves meet.

For instance, if you plug the number 3 into both of these functions, will you get the same answer?

$3\to 2\sqrt{x}\to 2\sqrt{3}\approx 3\text{.}\text{46}$

$3\to x+\frac{1}{2}\to 3\text{.}5$

Pretty close! Similarly, $2\sqrt{\text{.}1}\approx 0\text{.}\text{632}$ , which is quite close to 0.6. But if we want more exact answers, we will need to draw a much more exact graph, which becomes very time-consuming. (Rounded to three decimal places, the actual answers are 0.086 and 2.914.)

For more exact answers, we use analytic methods. Two such methods will be discussed in this chapter: substitution and elimination . A third method will be discussed in the section on Matrices.