Non-linear Systems of Equations, Linear Inequalities
define non-linear systems of equations. A non-linear system of equations is a
system where at least one of the equations is not linear.
solve non-linear systems of equations by substitution. Generally, we solve nonlinear systems using substitution or elimination. Solution sets may be
empty, finite (one or more), or infinite. Find all ordered pairs ( x, y ) such
that 9x + 3y ! 2 = 0 and 9x 2 + 3y 2 ! 7x = 0 using substitution.
solve non-linear systems of equations by elimination. Find all ordered pairs ( x, y )
such that 9x + 3y ! 2 = 0 and 9x 2 + 3y 2 ! 7x = 0 using elimination.
estimate solutions of non-linear systems of equations by graphing. Some nonlinear systems do not avail themselves to algebraic solution methods. In
those cases, we can only estimate the solutions graphically. Estimate the
x
solutions of the system: y =
and y = x 4 ! x . This system is
2
x +1
impossible to solve exactly. So we estimate the solutions by finding
(approximately) the points of intersection of the two graphs.
review graphing linear inequalities (test points). To graph a linear inequality, we
first graph the line. To graph the linear inequality !x + 3y < 12 , we first
graph the line !x + 3y = 12 . This line divides the plane into three areas:
points above the line, points on the line, and points below the line.
Algebraically, these three planar regions correspond to !x + 3y > 12 ,
!x + 3y = 12 , and !x + 3y < 12 , respectively. To match the correct
inequality with each half-plane, choose a test point in the half-plane. E.g.,
the origin lies in the half-plane below the line !x + 3y = 12 and also
satisfies the inequality !x + 3y < 12 . So the half-plane below the line
corresponds to the inequality !x + 3y < 12 .
graph system of linear inequalities (solution, vertices of solution set). Systems of
linear inequalities are studied in business and economics, particularly in
the study of linear programming. To graph the system:
!x + 3y " 12
x+ y #8
x "0
We begin by graphing the corresponding equations:
!x + 3y = 12; x + y = 8; x = 0 . We then determine the half-plane that
satisfies each inequality. We can now sketch the solution area. It is
important to determine the vertices of the solution area. A vertex is the
point of intersection of a pair of graphs. E.g., the lower left vertex is the
intersection of the graphs x = 0 and ! x + 3y = 12 , i.e., or (0, 4). The
upper left vertex is the intersection of the graphs x + y = 8 and x = 0 , or
(0, 8). The right-hand vertex is the intersection of the graphs
x + y = 8 and ! x + 3y = 12 . This can be solved by adding the two
equations together to find 4 y = 20 or y = 5 , and by substituting to find
x = 3 . So the intersection is (3, 5).