When we have two equations in two variables and are trying to find a
solution which fits both, we often solve one equation for one
of the variables and substitute into the second equation.
Note that if we have two unknowns (two variables) which we are trying
to find, we must have two equations--that is, two relationships
between the unknowns. Similarly, three unknowns requires three
equations, and so forth. The group of equations is known as a
system of equations, and we solve by finding the
simultaneous solutions to all of the equations in the system.
- Solve for x and y:
Solving the first equation for y gives y =
3 - (x/2), so, substituting for y in the
second equation we get
So the solution that solves both equations simultaneously is
- Solve for x and y:
Plugging in the expression x-1 for y in the second
equation, we get
Thus x=2 or x=-1. If x=2, then
y = 2-1 = 1, and if x=-1, then
y = 2-(-1) = 3. Thus the solutions to the
system are x=2 and y=1, or x=-1 and
- Solve for Q and a:
Solve the first equation for Q:
Then, plugging this in to the second equation,
Note that in this case we have found an approximate solution to
the system of equations:
- Find the points of intersection for the graphs in each of the
In both cases we solve the equations simultaneously by setting
the y-values equal to one another. For the left graph,
we are solving the system of equations
and thus x=2 or x=-1. To get the corresponding
y values we can use either equation. For x=2, we
get y=3, and for x=-1, y=0. Therefore the
points of intersection are (2,3) and (-1,0).
For the second graph, we are solving
And so want
However, we cannot use algebraic techniques to solve this
because the variable x appears both in the exponential
and on the left-hand side. We can guess one solution, however,
by noting that
so that x=2, y=3 is a solution to the system.
There is a second solution, as shown in the graph, which we can
estimate by tracing the graphs on a calculator. This is
Therefore the solutions are
Be sure you've gone through each step in the examples above before
doing these. Once you've worked them until you're sure that you
understand them, go on to the next section. There are more problems
of this type in the section test at the end of the section.
Note that you can get new practice problems by clicking the "Refresh"
button at the bottom of the practice set.
precal: 7.6 - solving simultaneous eqns
page created: Wed May 22 00:13:22 2013
©2003-2007 Gavin LaRose, Pat Shure /
University of Michigan Math Dept. /
Regents of the University of Michigan