Consider the equation
y=2xy=2x size 12{y=2 sqrt {x} } {}. How many
(x,y)(x,y) size 12{ \( x,y \) } {} pairs are there that satisfy this equation? Answer:
(0,0)(0,0) size 12{ \( 0,0 \) } {},
(1,2)(1,2) size 12{ \( 1,2 \) } {},
(4,4)(4,4) size 12{ \( 4,4 \) } {}, and
(9,6)(9,6) size 12{ \( 9,6 \) } {} are all solutions; and there is an *infinite number of other solutions*. (And don’t forget non-integer solutions, such as
(,1)(,1) size 12{ \( { size 8{1} } wideslash { size 8{4} } ,1 \) } {}!)

Now, consider the equation
y=x+12y=x+12 size 12{y=x+ { { size 8{1} } over { size 8{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)(x,y) size 12{ \( x,y \) } {} pairs make *both equations true*? To express the same question in terms of functions: what values can you hand the functions
2x2x size 12{2 sqrt {x} } {} and
x+12x+12 size 12{x+ { { size 8{1} } over { size 8{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=2xy=2x size 12{y=2 sqrt {x} } {} is the set of all points that satisfy that relationship; and the graph of
y=x+12y=x+12 size 12{y=x+ { { size 8{1} } over { size 8{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+by=mx+b size 12{y= ital "mx"+b} {} format. The
yy size 12{y} {}-intercept is
1212
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=xy=x size 12{y= sqrt {x} } {} looks like, you can stretch the graph vertically to get
y=2xy=2x size 12{y=2 sqrt {x} } {}, since all the
yy size 12{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
xx size 12{x} {} is barely greater than 0 (say,
x=0.1x=0.1 size 12{x=0 "." 1} {}), and the other occurs at approximately
x=3x=3 size 12{x=3} {}. There will be no more points of intersection after this, because the line will rise faster than the curve.

y
=
2
x
y
=
2
x
size 12{y=2 sqrt {x} } {}

y
=
x
+
1
2
y
=
x
+
1
2
size 12{y=x+ { { size 8{1} } over { size 8{2} } } } {}

From graphing...

x=0.1x=0.1 size 12{x=0 "." 1} {},
x=3x=3 size 12{x=3} {}

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
→
2
x
→
2
3
≈
3
.
46
3
→
2
x
→
2
3
≈
3
.
46
size 12{3 rightarrow 2 sqrt {x} rightarrow 2 sqrt {3} approx 3 "." "46"} {}

3
→
x
+
1
2
→
3
.
5
3
→
x
+
1
2
→
3
.
5
size 12{3 rightarrow x+ { { size 8{1} } over { size 8{2} } } rightarrow 3 "." 5} {}

Pretty close! Similarly,
2.1≈0.6322.1≈0.632 size 12{2 sqrt { "." 1} approx 0 "." "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.

Comments:"DAISY and BRF versions of this collection are available."