Chapter Eight — Systems of Linear Equations and Inequalities
The Humongous Book of Algebra Problems
173
Figure 8-10: The shaded region represents the solution to the system of inequalities.
Note that the individual inequality graphs, and their accompanying lightly
shaded regions, are omitted from Figure 8-10. When a system consists of three
or more inequalities, the value of the individual graphs is lessened because the
graph begins to look cluttered. If the graph of a system is bounded by multiple
inequalities, ensure that you clearly indicate which region constitutes the
solution.
Linear Programming
Use the sharp points at the edge of a shaded region
Note: Problems 8.36–8.39 refer to the expression P = 6x – 2y and the following system of
linear equations.
8.36 Graph the system of inequalities.
Apply the method described in Problems 8.29–8.35 to generate the graph, as
illustrated by Figure 8-11.
In Problems
8.36–8.39,
you’ll nd the
highest and lowest
possible values of P
given the limitations
dened by the system
of inequalities. In other
words, you’ll nd out
what point from the
shaded region of the
graph, when plugged
into P = 6x – 2y, makes
P the biggest and
which point makes
P the smallest.
Chapter Eight — Systems of Linear Equations and Inequalities
The Humongous Book of Algebra Problems
174
Figure 8-11: All the points in the shaded region and the boundaries of the region
represent solutions to the system of inequalities.
Note: Problems 8.36–8.39 refer to the expression P = 6x – 2y and the system of linear
inequalities identified in Problem 8.36.
8.37 Consider the region graphed in Problem 8.36. Identify the points that could
generate an optimal value of P.
The optimal (maximum and minimum) values of P will occur at one of the
vertices of the solution region, which are identified as points A, B, C, and D in
Figure 8-12.
Figure 8-12: Points A, B, C, and D are the vertices of the feasible solution set.
The boun-
daries are also
solutions because
all the inequalities
contain either “≤”
or “≥.”
The vertices
are the intersection
points of the boundary
lines along the edge of
the region, basically
the “corners” of the
shaded region.
The shaded
region of the
graph is sometimes
referred to as the set
of “feasible solutions” in
linear programming
problems.
Chapter Eight — Systems of Linear Equations and Inequalities
The Humongous Book of Algebra Problems
175
Point D is the origin: D = (0,0). Point A is the y-intercept of the line .
The line is in slope-intercept form, so the y-intercept is 3, the constant in the
equation. Therefore, A = (0,3).
Point C is the x-intercept of the line Substitute y = 0 into the
equation and solve for x to calculate the x-intercept.
Therefore, .
Point B is the intersection of lines and . Solve the
system containing those two equations using substitution to identify B.
Multiply every term by 4 to eliminate the fractions.
Solve for x.
Substitute x = 2 into either of the intersecting linear equations to determine the
corresponding value of y.
Both of
the equations
are solved for y, so
substituting
into
(or
vice versa) ends
up being the x-
expressions set
equal to each
other.
Chapter Eight — Systems of Linear Equations and Inequalities
The Humongous Book of Algebra Problems
176
The graphs of and intersect at .
Note: Problems 8.36–8.39 refer to the expression P = 6x – 2y and the system of linear
inequalities identified in Problem 8.36.
8.38 Evaluate P at each of the coordinate pairs identified in Problem 8.37.
Substitute A = (0,3), , , and D = (0,0) into P = 6x – 2y.
Note: Problems 8.36–8.39 refer to the expression P = 6x – 2y and the system of linear
inequalities identified in Problem 8.36.
8.39 Identify the maximum and minimum values of P given the constraints defined
by the system of linear inequalities.
According to Problem 8.38, the maximum value of P is , occurring when
and y = 0. The minimum value of P is –6, occurring when x = 0 and
y = –3.
The optimal
(maximum or
minimum) values
of P will always
correspond with
a vertex of the
graph. No other
coordinate pair
in the shaded
region will result
in a higher or
lower value
of P.
Chapter Eight — Systems of Linear Equations and Inequalities
The Humongous Book of Algebra Problems
177
Note: Problems 8.40–8.43 refer to the expression S = x + 5y and the below system of linear
equations.
8.40 Graph the system of inequalities.
Apply the method described in Problems 8.29–8.35 to generate the graph in
Figure 8-13.
Figure 8-13: The shaded region of the coordinate plane is the set of feasible solutions,
coordinates that comply with the constraints defined by the system of
inequalities.
The inequal-
ities of the system
are sometimes called
the “constraints” in a
linear programming
problem.
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