Unit 2: Systems
2015-2016: Algebra II CP
Define
 Two or more equations with the same variables
System of Equations
Example
5 x  2 y

3x  y
3x  2 y  5

5 x  3 y  7
Non Example
5x  6 y  9
3x  2 y  6

5 z  2 y  8
Solving a Systems
MAKES TRUE
When solving a system with two variables, we are trying to find an ordered pair that satisfies all of the equations.
Is the ordered pair a solution of the system?
(2, -4)
3𝑥 + 2𝑦 = −2
{
}
−4𝑥 + 5𝑦 = −28
There are three methods to solving systems of equations.
(1) Graphing
𝑥−𝑦 =5
{
}
𝑥 + 2𝑦 = −4
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Unit 2: Systems
METHOD 2 SUBSTITUTION
2015-2016: Algebra II CP
Equation 1:
x
-
y
Equation 2:
x
+
2y =
=
5
-4
(1) Get x or y alone for one of the equations.
(2) Substitute BLOB into the OTHER equation then SOLVE.
(3) Plug answer back into BLOB
(4) Write answer as ordered pair.
(5) Check your answer.
(6) Write your final solution to the problem.
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Unit 2: Systems
2015-2016: Algebra II CP
METHOD 3 ELIMINATION
(1)
Look for opposites (What will you eliminate x or y?)
x or y
Multiply by…
#1
x
-
y
#2
x
+
2y =
(2)
Add Equations and Solve
(3)
Plug answer back into one of the original equations to find the other variables
(4)
Write answer as ordered pair
(5)
Check your answer
(6)
Write your final answer
=
5
-4
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Unit 2: Systems
2015-2016: Algebra II CP
We classify systems by the number of solutions.
CONSISTENT: the system has at least one solution
 INDEPENDENT: if there is EXACTLY one solution (intersecting lines)
 DEPENDENT: infinite number of solutions (same line)
INCONSISTENT: the system has no solution (parallel lines)
One Solutions
Many Solutions
No Solutions
Consistent Independent
Consistent Dependent
Inconsistent
Intersecting lines
Same line
Parallel lines
(x , y)
True statement
False statement
EXAMPLE: Solve the system of equations.
(1.)
Graph the system x + y = 3 and -2x + y = -6.
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Unit 2: Systems
2015-2016: Algebra II CP
EXAMPLE: Solve the system of equations.
(2.)
8x + y = 27 and -3x + 4y = 3
(3.)
6w = 12 – 4x and 6x = -9w + 18
Page 5 of 16
Unit 2: Systems
2015-2016: Algebra II CP
Define
 A set of inequalities with the same variables.
 Two or more
  solid line
SYSTEM OF INEQUALITIES
(4.)
 > dashed line
 y  2x  4
 y  0.5 x  3
Solve the system 
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Unit 2: Systems
(5.)
 y  2 x  5

Solve the system 
1
 y   4 x  3
(6.)
y  x

Solve the system 
4
y  x 5
3

2015-2016: Algebra II CP
Page 7 of 16
Unit 2: Systems
2015-2016: Algebra II CP
y  x  5
y  x  4
(7.)
Solve the system 
(8.)
Solve the system 
 y  2 x
 y  3
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Unit 2: Systems
(9.)
2015-2016: Algebra II CP
Chelsea has final exams in calculus, physics, and history. She has up to 25 hours to study for exams. She plans
to study history for 2 hours. She needs to spend at least 7 hours studying for calculus, but over 14 is too much.
She hopes to spend between 8 and 12 hours on physics. Write and graph a system of inequalities to represent
the situation. (x-axis 2’s and y-axis by 2’s)
Page 9 of 16
Unit 2: Systems
2015-2016: Algebra II CP
(10.) Bob and Jane are driving across the country with their two children. They plan on driving a maximum of 10
hours each day. Bob wants to drive at least 4 hours a day but no more than 8 hours a day. Jane can drive
between 2 and 5 hours per day. Write and graph a system of inequalities that represent this information. (x-axis
by 1’s and y-axis by 1’s)
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Unit 2: Systems
Define




2015-2016: Algebra II CP
LINEAR PROGRAMMING
Method for finding minimum and maximum values of a function
A set of inequalities (these are the constraints)
The graph is the feasible region
Plug in the vertices of the feasible region to determine minimum or maximum values
EXAMPLES
(11.) Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the
maximum and minimum values of the function for this region.
3 y 6
y  3 x  12
y  2 x  6
f ( x, y )  4 x  2 y
Page 11 of 16
Unit 2: Systems
2015-2016: Algebra II CP
EXAMPLES
(12.) Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the
maximum and minimum values of the function for this region.
2  x  6
1 y  5
y  x3
f ( x, y )  5 x  2 y
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Unit 2: Systems
2015-2016: Algebra II CP
EXAMPLES
(13.) Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the
maximum and minimum values of the function for this region.
6  y  2
y  x  3
y  x2 7
f ( x, y )  6 x  4 y
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Unit 2: Systems
2015-2016: Algebra II CP
EXAMPLES
(14.) Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the
maximum and minimum values of the function for this region.
y 8
y  x 4
y   x  10
f ( x, y )  6 x  8 y
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Unit 2: Systems
Define
2015-2016: Algebra II CP
OPTIMIZE
 To seek best price of amount to minimize costs or maximize profits
(15.) An electronic company produces digital audio players and phones. A sign on the company bulletin is shown.
If at least 2000 items must be produced per shift, how many of each type should be made to minimize costs?
The company is experiencing limitations, or constraints, on production caused by customer demand, shipping, and
the productivity of their factory. A system of inequalities can be used to represent these constraints.
(x-axis and y-axis by 200)
Our Goal: Production Per Shift
Unit
Minimum Maximum Cost per Unit
Audio 600
1500
$55
Phone 800
1700
$95
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Unit 2: Systems
2015-2016: Algebra II CP
(16.) Each week, Mac can make 10 to 25 necklaces and 15 to 40 pairs of earrings. If she earns profits of $3 on
each pair of earrings and $5 on each necklace, and she plans to sell at least 30 pieces of jewelry, how can she
maximize profits? (x-axis and y-axis by 2’s)
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