Chapter 6
Systems of
Equations and
Inequalities
6.1 Linear Systems in Two Variables with Applications
Solving systems Graphically
4 x  3 y  9

 2 x  y  5
The solution is
(3,1)
6.1 Linear Systems in Two Variables with Applications
Solving systems by substitution
1. Solve one of the equations for x in terms of y or y
in terms of x
2. Substitute for the appropriate variable in the other
equation
3. Solve the resulting equation for the remaining
variable. (This will give you one coordinate of the
ordered pair solution.)
4. Substitute the value from step 3 back into either
of the original equations to determine the value of
the other coordinate.
5. Write the answer as an ordered pair and check.
6.1 Linear Systems in Two Variables with Applications
6.1 Linear Systems in Two Variables with Applications
1.
2.
3.
4.
5.
Solve one of the equations for x in
terms of y or y in terms of x
Substitute for the appropriate
variable in the other equation
Solve the resulting equation for the
remaining variable. (This will give
you one coordinate of the ordered
pair solution.)
Substitute the value from step 3
back into either of the original
equations to determine the value of
the other coordinate.
Write the answer as an ordered
pair and check.
6.1 Linear Systems in Two Variables with Applications
6.1 Linear Systems in Two Variables with Applications
Solving Systems Using Elimination
1. Be sure each equation is in standard form: Ax+By=C
2. If desired or needed, clear fractions or decimals using a
multiplier.
3. Multiply one or both equations by a number that will create
coefficients of x (or y) that are additive inverses.
4. Combine the two equations using vertical addition.
5. Solve the resulting equation for the remaining variable.
6. Substitute this x- or y-value back into either of the original
equations and solve for the other unknown.
7. Write the answer as an ordered pair and check the solution
in both original equations.
6.1 Linear Systems in Two Variables with Applications
1.
2.
3.
4.
5.
6.
7.
Be sure each equation is in standard form:
Ax+By=C
If desired or needed, clear fractions or decimals
using a multiplier.
Multiply one or both equations by a number that will
create coefficients of x (or y) that are additive
inverses.
Combine the two equations using vertical addition.
Solve the resulting equation for the remaining
variable.
Substitute this x- or y-value back into either of the
original equations and solve for the other unknown.
Write the answer as an ordered pair and check the
solution in both original equations.
6.1 Linear Systems in Two Variables with Applications
6.1 Linear Systems in Two Variables with Applications
Homework pg 573 1-74
6.2 Linear Systems in Three Variables with Applications
Transformations of a Linear System
1. You can change the order the equations are written.
2. You can multiply both sides of an equation by any
nonzero constant.
3. You can add two equations in the system and use the
result to replace any other equation in the system.
6.2 Linear Systems in Three Variables with Applications
1.
2 x  y  3 z  3

3 x  2 y  4 z  2
4 x  2 y  6 z  7

2.
3.
You can change the order the
equations are written.
You can multiply both sides of an
equation by any nonzero constant.
You can add two equations in the
system and use the result to
replace any other equation in the
system.
6.2 Linear Systems in Three Variables with Applications
1.
x  2 y  z  1

x  z  3
2 x  y  z  3

2.
3.
You can change the order the
equations are written.
You can multiply both sides of an
equation by any nonzero constant.
You can add two equations in the
system and use the result to
replace any other equation in the
system.
6.2 Linear Systems in Three Variables with Applications
1.
A  2B  5

 B  3C  7
2 A  B  C  1

2.
3.
You can change the order the
equations are written.
You can multiply both sides of an
equation by any nonzero constant.
You can add two equations in the
system and use the result to
replace any other equation in the
system.
6.2 Linear Systems in Three Variables with Applications
Homework pg 586 1-60
6.3 Systems of Inequalities and Linear Programming
Solving a Linear Inequality
1. Graph the boundary line by solving for y and using the slope-intercept
form.
• Use a solid line if the boundary is included in the solution set.
• Use a dashed line if the boundary is excluded from the solution set.
2. For “greater than” inequalities shade the upper half plane. For “less
than” inequalities shade the lower half plane.
3. Select a test point from the shaded solution region and substitute the
x- and y-values into the original inequality to verify the correct region
shaded.
6.3 Systems of Inequalities and Linear Programming
1.
2.
3.
Graph the boundary line by solving for y and
using the slope-intercept form.
•
Use a solid line if the boundary is included
in the solution set.
•
Use a dashed line if the boundary is
excluded from the solution set.
For “greater than” inequalities shade the upper
half plane. For “less than” inequalities shade the
lower half plane.
Select a test point from the shaded solution
region and substitute the x- and y-values into
the original inequality to verify the correct region
shaded.
6.3 Systems of Inequalities and Linear Programming
Solving Linear Programming Applications
1. Identify the main objective and the decision variables (descriptive
variables may help.)
2. Write the objective function in terms of these variables.
3. Organize all information in a table, with the decision variables and
constraints heading up the columns, and their components leading
each row. Complete that table using the information given and
write the constraint inequalities using the decision variables,
constraints, and the domain.
4. Graph the constraint inequalities and determine the feasible
region.
5. Identify all corner points of the feasible region and test these
points in the objective function to determine the optimal
solution(s).
6.3 Systems of Inequalities and Linear Programming
A calculator company produces a scientific calculator and a graphing
calculator. Long-term projections indicate an expected demand of at least
100 scientific and 80 graphing calculators each day. Because of limitations
on production capacity, no more than 200 scientific and 170 graphing
calculators can be made daily. To satisfy a shipping contract, a total of at
least 200 calculators much be shipped each day.
If each scientific calculator sold results in a $2 loss, but each graphing
calculator produces a $5 profit, how many of each type should be made
daily to maximize net profits?
 2 S  5 G  Profit
S is # of scientific
calculators produced
S  100
G  80
G is # of graphing
calculators produced
S  200
G  170
G  S  200
6.3 Systems of Inequalities and Linear Programming
S  100
G
G  80
S  200
G  170
G  S  200
S
6.3 Systems of Inequalities and Linear Programming
S  100
G  80
S  200
G
(100,170)
(200,170)
G  170
G  S  200
(100,100)
(120,80)
(200,80)
S
6.3 Systems of Inequalities and Linear Programming
 2 S  5 G  Profit
(100,170)
$850
(200,170)
$450
(100,100)
(120,80)
$300
$160
(200,80)
$0
6.3 Systems of Inequalities and Linear Programming
Homework pg 600 1-60
6.7 Solving Linear Systems Using Matrix Equations
Chapter 6 Review