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1
Unit 2 Systems of Equations &
Inequalities
General Outcome: • Develop algebraic and graphical reasoning through the study of relations.
Specific Outcomes:
2.1 Solve, algebraically and graphically, problems that involve systems of
linear-quadratic and quadratic-quadratic equations in two variables.
2.2 Solve problems that involve linear and quadratic inequalities in two variables.
2.3 Solve problems that involve quadratic inequalities in one variable.
Topics
• Linear Systems of Equations Review Page 2
• Solving by Graphing (Outcome 2.1) Page 13
• Solving by Substitution (Outcome 2.1) Page 20
• Solving by Elimination (Outcome 2.1) Page 29
• Graphing Inequalities in One Variable (Outcome 2.2) Page 38
• Solving Quadratic Inequalities with (Outcome 2.3) Page 48
One Variable
• Solving Quadratic Inequalities with (Outcome 2.2) Page 55
Two Variables
2
Unit 2 Systems of Equations &
Inequalities
Review of Linear Systems of Equations:
Systems of Equations:
A system of equations involves 2 or more equations that are
considered at the same time.
Ex) Consider the system given by the following equations:
2 1
3 9
y x
y x
= −
= − +
• Graph 2 1y x= −
• Graph 3 9y x= − +
• Determine the solution to the system given by 2 1
3 9
y x
y x
= −
= − +
3
Solving Systems Graphically:
Ex) Solve the following systems of equations by graphing each
system.
2 4y x= −
3 2 4x y− =
Solving Systems by Graphing on the Calculator:
• Solve each equation for y.
• Enter equations into 1y and 2y , then graph the equations.
The point of intersection must be visible.
• Determine the point of intersection using the intersect
feature.
4
Ex) Solve the following systems of equations by graphing (use
your graphing calculator).
a) 2 5x y+ =
2 10x y− =
b) 5 3 6x y+ =
20 24 12x y= −
5
Solving by Substitution:
• Solve for one variable in one of the equations (choose the
one that is easiest to solve for).
• Substitute this expression into the other equation.
• Solve the single variable equation now created (solve for
the 1st variable).
• Solve for the second variable by substituting the value
already solved for into one of the equations.
Ex) Solve the following systems using the method of
substitution.
a) 5 3 2 0x y− − =
7 0x y+ =
6
b) 4 6x y+ =
2 3 1x y− =
c) 2 9x y− + =
5 5 0x y+ + =
7
Solving by Elimination:
• Arrange the equations so that one is above the other and
corresponding terms and the “=” sign are aligned.
• Manipulate equations so the absolute value of the
numerical coefficients of one pair of like terms are the
same.
• Either add or subtract the equations to eliminate one of the
variables.
• Solve for the 1st variable.
• Solve for the second variable by substituting the value
already solved for into one of the equations.
Ex) Solve the following systems of equations using the method
of elimination.
a) 3 2 19x y+ = , 5 2 5x y− =
8
b) 7 25x y+ = −
5 13 7x y+ = −
c) 4 2 16x y− = −
8 3 46x y+ = −
d) 4
42 3
x y+ =
3 5 6x y+ =
9
Linear Systems of Equations Review Assignment:
1) Solve the following systems of equations by the method of graphing.
a) 5y x= − b) 2 2m n+ =
3y x= − 3 2 6m n+ = −
c) 4 0x y+ − = d) 2 4x y− = −
5 8 0x y− − = 2 6x y+ =
2) Solve the following systems by the method of substitution.
a) 6 2y x= −
3 2 10x y+ =
10
b) 3 2 0x y+ − =
5 2 3 0x y+ − =
c) 7 2b a= −
4 a b= +
d) 3 6 1m n− =
3 2m n+ =
11
3) Solve the following systems using the method of elimination.
a) 2 3 4x y+ =
4 3 10x y− = −
b) 4 5 3a b+ = −
4 9 1a b+ =
c) 3 4 17x y+ =
7 2 17x y− =
12
d) 2 5 3x y− =
3 2 14x y+ =
4) Solve each system algebraically (use substitution or elimination).
a) 33 2
x y+ =
3 1
42 5
x y+ ++ =
b) 0.5 0.4 0.5x y− =
3 0.8 1.4x y+ =
13
Non-Linear Systems:
In this unit we will now consider systems of equations made up
of a linear and a quadratic equation or systems made of two
quadratic equations.
Possible Number of Solutions:
Linear-Quadratic
Quadratic-Quadratic
14
Steps for Solving Graphically:
• Solve each equation for y.
• Enter equations into 1y and 2y , then graph the equations.
The point(s) of intersection must be visible.
• Determine the point(s) of intersection using the intersect
feature.
Ex) Solve the following systems graphically. Round to the
nearest hundred if necessary.
a) 4 3 0x y− + = 22 8 3 0x x y+ − + =
b) 22 16 35x x y− − = − 22 8 11x x y− − = −
c) ( )21 6 18
3y x−= − +
( )2
2 14 16y x= − − +
15
Ex) During a stunt, two Cirque du Soleil performers are
launched toward each other from two slightly offset
seesaws. The first performer is launched, and 1 second later
the second performer is launched in the opposite direction.
They both perform a flip and give each other a high five in
the air. Each performer is in the air for 2 seconds. The
height above the seesaw versus time for each performer
during the stunt is approximated by a parabola as shown.
a) Determine the system of equation that models the
performers’ height during the stunt.
b) Solve the system graphically using your calculator.
c) Interpret your solution with respect to this situation.
16
Solving Systems Graphically Assignment:
1) Verify that ( )0, 5− and ( )3, 2− are solutions to the following system of
equations.
2 4 5y x x= − + −
5y x= −
2) Solve each system by graphing.
a) 7y x= + b) ( ) 5f x x= − +
( )2
2 3y x= + + ( )21
( ) 4 12
g x x= − +
c) 2 16 59x x y+ + = − d) 2 3 0x y+ − =
2 60x y− = 2 1 0x y− + =
17
e) 2 10 32y x x= − + f) 2 16 60a b b= − + 22 32 137y x x= − + 12 55a b= −
g) 23 12 17y x x= − + h) 22 20 40m m n+ + = − 20.25 0.5 1.75y x x= − + + 5 2 26 0m n+ + =
i) 2 2 2 7 0a a b+ − − = j) 20 40 400x x y= + − +
23 12 6 0a a b+ − + = 2 30 225x y x= + −
18
3) For each situation, sketch a graph that represents a quadratic – quadratic
system of equations with two real solutions, so that the two parabolas have
a) the same axis of symmetry b) the same axis of symmetry and
the same y-intercept
c) different axis of symmetry and d) the same x-intercepts
the same y-intercept\
4) Vertical curves are used in the construction of roller coasters. One downward-
sloping grade line, modelled by the equation 0.04 3.9y x= − + , is followed by
an upward-sloping grade line modelled by the equation 0.03 2.675y x= + . The
vertical curve between the two lines is modelled by the equation 20.001 0.04 3.9y x x= − + . Determine the coordinates of the beginning and the
end of the curve.
19
5) A frog jumps to catch a grasshopper. The frog reaches a maximum height of 25
cm and travels a horizontal distance of 100 cm. A grasshopper, located 30 cm
in front of the frog, starts to jump at the same time as the frog. The grasshopper
reaches a maximum height of 36 cm and travels a horizontal distance of 48 cm.
The frog and the grasshopper both jump in the same direction.
a) Consider the frog’s starting position to be at the origin of a coordinate grid.
Sketch a diagram to model the given information.
b) Determine a quadratic equation to model the frog’s height compared to the
horizontal distance it travelled and a quadratic equation to model the
grasshopper’s height compared to the horizontal distance traveled.
c) Solve the system of two equations.
d) What does this solution represent in the context of this question?
20
Solving by Substitution:
Linear-Quadratic System:
o Solve for y in the linear equation.
o Substitute this expression into the quadratic equation.
o Solve the quadratic equation by combining like terms
and using the method of factoring, completing the
square, or quadratic formula.
o Solve for the y-variable by substituting the x-value(s)
already solved for into one of the equations.
Quadratic-Quadratic System:
o Solve one of the equations for y.
o Substitute this expression in for y in the other
equation.
o Solve the quadratic equation by combining like terms
and using the method of factoring, completing the
square, or quadratic formula.
o Solve for the y-variable by substituting the x-value(s)
already solved for into one of the equations.
Ex) Solve the following systems using the method of
substitution.
a) 5 10x y− = 2 2 0x x y+ − =
21
b) 23 24 41y x x= + −
6 17 0x y− + =
c) 2 2 40x y+ =
3 12 0x y+ − =
d) 23 2 0x x y− − − = 26 4 4x x y+ − =
22
e) 2 6 28y x x= + − 2 12 34 3x x y+ = − −
f) 24 28 12 36y x x+ = − + 265 2 24y x x− = −
23
Solving by Substitution Assignment:
1) Solve the following systems using the method of substitution.
a) 2 2 0x y− + =
4 14x y= −
b) 2 8 19x y x+ = +
2 7 11x y x− = −
24
c) 27 5 8 0a a b+ − − =
10 2 40a a− = −
d) 22 4 2 6p p m= − +
25 8 10 5m p p+ = +
25
e) 24 8 6h t− =
26 9 12h t− =
f) 7
18
y x−
− =
23 8 1x y x+ = −
26
2) Determine the values of m and n if ( )2, 8 is a solution to the following system
of equations.
2 16mx y− = 2 2mx y n+ =
3) Two integers have a difference of 30− . When the larger integer is increased by
3 and added to the square of the smaller integer, the result is 189. Determine
the value of the integers.
27
4) A stuntman skies off the edge of mountain, free-falls for several seconds, and
then opens a parachute. The height, h, in meters, of the stuntman above the
ground t seconds after leaving the edge of the mountain is given by two
functions.
2( ) 4.9 2015h t t= − + represents the height of the stuntman before he opens the
parachute.
( ) 10.5 980h t t= − + represents the height of the stuntman after he opens the
parachute.
For how many seconds does the stuntman free-fall before he opens his
parachute, and what height above the ground was the stuntman when he
opened the parachute?
28
5) Solve the following system.
2 1xy
x
−=
2 02
xy
x+ − =
+
29
Solving by Elimination:
• Arrange the equations so that one is above the other and
corresponding terms and the “=” sign are aligned.
• Manipulate equations so the absolute value of the
numerical coefficients on the y-term are the same.
• Either add or subtract the equations to eliminate the
y-variables.
• Solve for the x-variable.
• Solve for the y-variable by substituting the value(s) of x
already solved for into one of the equations.
Ex) Solve the following systems using the method of
elimination.
a) 3 2 29x y+ = 24 10 13y x x= − +
30
b) 214 14y x x+ + = −
5 76y x= +
c) 24 5 6y x x− = −
2 16 3 9y x− = +
31
d) 23 36 75x x y− + = − 23 54 189x x y− − = −
e) ( )2
6 7 3y x= − +
( )2
3 13 9y x= − − +
32
f) 24 4 13x x y− = + 220 20 100y x x+ = − +
33
Solving by Elimination Assignment:
1) Solve the following systems using the method of substitution.
a) 22 4 3x x y− + =
4 2 7x y− = −
b) 26 3 2 5x x y− = −
22 4x x y+ = −
34
c) 23 4 8 0x x y+ − − =
23 2 4y x x+ = +
d) 29 8 14w k+ = −
2 2w k+ = −
35
e) 22 6y x x+ = −
23 2x y x+ − =
f) 2 48 1 1
09 3 3
x x y− + + =
25 3 1 1
04 2 4 2
x x y−
− + − =
36
2) The perimeter of the right triangle shown below is 60 m. The area of the
triangle is 10y square meters. Solve for x and y.
14y +
2x
5 1x −
3) The number of centimeters in the circumference of a circle is three times the
number of square centimeters in the area of the circle. Determine the radius,
circumference, and area of the circle with this property.
37
4) Kate is an industrial design engineer. She is creating the program for cutting
fabric for a shade sail. The shape of a shade sail is defined by three
intersecting parabolas. The equations of the parabolas are given by
2 8 16y x x= + + 2 8 16y x x= − +
2
28
xy
−= +
a) Determine the coordinates of the three vertices of the sail
b) Estimate the area of the material required to make the sail.
38
Graphing Linear Inequalities in 2 Variables:
As we have seen linear inequalities in one variable can be
graphed on a number line (one dimension). Linear inequalities in
two variables must be graphed on a coordinate plane (two
dimensions).
Ex) Graph the inequality given below.
2 2
3y x +
• Graph the line 2 23
y x= +
• Because we want 2 23
y x +
(y is less than 2 23
x + ) shade
the region below the line
• Check your answer (check
with a point in the shaded
region).
How do we distinguish between 2 1y x + and 2 1y x + ?
2 1y x→ + is graphed with a dotted line ------------
2 1y x→ + is graphed with a solid line
39
Ex) Graph each of the following inequalities. Check your
solutions.
a) 4 3 12 27x y− +
b) 4x c) 5y −
40
Inequalities with Restrictions:
Restrictions limit the region to be shaded.
Ex) Graph the following inequalities with restrictions.
a) 2 53
y x − , 0
restrictions0
x
y
b) 2 5 13x y− , 0
0
x
y
41
Ex) Mary is buying flowers for her yard. During a sale she can
buy a flat o f marigolds for $5 and a flat of petunias for $6.
If she has $60 to spend, how many of each type of flower
can she buy. Graph your results.
42
Graphing Linear Inequalities in Two Variables Assignment:
1) For each inequality, state whether or not the given points are solutions to the
inequality.
a) 3y x +
( )7, 10 ( )7, 10− ( )6, 7 ( )12, 9
b) 5x y− + −
( )2, 3 ( )6, 12− − ( )4, 1− ( )8, 2−
c) 3 2 12x y−
( )6, 3 ( )12, 4− ( )6, 3− − ( )5, 1
2) Graph each inequality.
a) 2 5y x − +
43
b) 3 8y x−
c) 4 2 12 0x y+ −
d) 4 10 40x y−
44
e) 6x y −
f) 6 3 21x y+
g) 10 2.5x y
45
h) 2.5 10x y
i) 0.8 0.4 0x y−
3) Determine the inequality that corresponds to each graph.
a)
46
b)
c)
d)
47
4) Amaruq has a part-time job that pays her $12/h. She also sews baby moccasins
and sells them for a profit of $12 each. Amaruq wants to earn at least
$250/week.
a) Write an inequality that represents the number of hours that Amaruq can
work and the number of baby moccasins she can sell to earn at least $250.
Include any restrictions on the variables.
b) Graph the inequality.
48
Solving Quadratic Inequalities with One Variable:
Solving Linear Inequalities with One Variable (Review):
Ex) Solve the following.
a) 5 4 2 23x x− = + b) 2 7 3 47y y− + − =
Solving Quadratic Inequalities:
• Bring all terms to one side of the inequality
• Graph the related quadratic function and make note of the
x-intercepts
o If function is less than 0, identify regions below the
x-axis
o If function is greater than 0, identify regions above the
x-axis
o If function can equal 0, include all points on the x-axis
• Make inequality statements that represent each region
49
Ex) Solve the inequality below given its graph.
Graph of 220 21y x x= − −
210 21 0x x− −
Ex) Solve the following inequalities.
a) ( )2
2 9x −
50
b) 2 2 8 0x x+ −
Ex) Solve the following inequalities.
a) 2 4 77 0x x− −
b) 2 5 37 4x x+ −
51
c) 22 3 27x x+
d) 2 18 30 7 6x x x+ + −
52
Solving Quadratic Inequalities with One Variable Assignment:
1) Solve the following inequalities.
a) ( )6 40x x + b) 2 14 24 0x x− − −
c) 26 11 35x x + d) 28 5 2x x+ −
e) 2 3 18x x+ f) 2 3 4x x+ −
53
g) 24 27 18 0x x− + h) 26 16x x− −
i) 2 10 16 0x x− + j) 212 11 15 0x x− −
k) 2 2 12 0x x− − l)
2 6 9 0x x− +
54
m) 2 3 6 10x x x− + n) 2 22 12 11 2 13x x x x+ − + +
2) Suppose that an engineer determines that she can use the formula 2 14t P− +
to estimate when the price of carbon fibre will be P dollars per kilogram or less
in t years from the present. When will the carbon fibre be available at $10/kg
or less?
55
Graphing Quadratic Inequalities with Two Variables:
Like linear inequalities with two variables, we will solve
quadratic inequalities with two variables by graphing a related
equation and shading the appropriate region.
Ex) Graph the inequality given below.
( )2
5 3y x − −
• Graph the parabola
( )2
5 3y x= − −
• Because we want
( )2
5 3y x − −
(y is greater than ( ) 25 3x − − )
shade the region above the
parabola
• Check your answer (check
with a point in the shaded
region).
*Remember
if y or y use a solid line
if y or y use a dotted line
56
Ex) Graph the following quadratic inequalities.
a) ( )2
2 3 1y x − − +
b) 2 2 3y x x − −
57
c) 2 4 5y x x − −
d) 2 23 14 4x y x+ −
58
Ex) Graph the following systems of inequalities.
a) 2( 3) 5y x + −
2
43
y x +
b) 2 2 6y x x − − +
2 3y x − +
59
Graphing Quadratic Inequalities with Two Variables Assignment:
1) Determine whether or not the ordered pairs are solutions to the given
inequality.
a) 2 3y x +
( )2, 6 ( )4, 20 ( )1, 3− ( )3, 12−
b) 2 3 4y x x − + −
( )2, 2− ( )4, 1− ( )0, 6− ( )2, 15− −
c) 21
52
y x x−
− +
( )4, 2− ( )1, 5− ( )1, 3.5 ( )3, 2.5
60
2) Write an inequality to describe each graph given below.
a)
b)
c)
d)
61
3) Graph each inequality.
a) ( )2
2 3 4y x + +
b) ( )21
4 12
y x−
− −
c) ( )2
3 1 5y x + +
62
d) ( )21
7 24
y x − −
e) 2 12 37y x x + +
f) 2 6y x x + −
63
g) 2 5 4y x x − +
h) 2 8 16y x x + +
4) Write an inequality to describe each graph.
a)
64
b)
5) In order to get the longest possible jump, ski jumpers need to have as much lift
area, L, in square meters, as possible while in the air. One of the many
variables that influences the amount of lift area is the hip angle, a, in degrees,
of the skier. The relationship between the two is given by 20.000125 0.040 2.442L a a − + − . What is the range of hip angles that will
generate lift area of at least 0.50 m?
65
6) In order to conduct microgravity research, the Canadian Space Agency uses a
Falcon 20 jet that flies a parabolic path. As the jet nears the vertex of the
parabola, the passengers in the jet experience nearly zero gravity that lasts for a
short period of time. The function 22.944 191.360 6950.400h t t= − + + models
the flight of a jet on a parabolic path for time, t, in seconds, since
weightlessness has been achieved and height, h, in meters. The passengers
begin to experience weightlessness when the jet climbs above 9600 m.
Determine the time period for which the jet is above 9600 m. For how long
does the microgravity exist on the flight?
7) Graph the following system of inequalities.
a) ( )2
2 1 5y x − − b) ( )2
1 5y x − −
1
13
y x − + 2( 3) 2y x − − +
66
Answers Linear Systems of Equations Review Assignment:
1. a) ( )4, 1− b) 4m = − , 3n = c) ( )2, 2 d) 1
, 52
2. a) ( )2, 2 b) ( )1, 1− c) 1a = − , 5b = d) 1m = , 1
3n =
3. a) ( )1, 2− b) 2a = − , 1b = c) ( )3, 2 d) ( )4, 1
4. a) ( )3, 4 b) 3 1
, 5 2
−
Solving Systems Graphically Assignment:
2. a) ( )3, 4− and ( )0, 7 b) ( )2, 3 and ( )4, 1
c) ( )14.5, 37.25− − and ( )2, 31− − d) ( )1, 0− and ( )1, 0
e) ( )7, 11 and ( )15, 107 f) ( )5, 5 and ( )221, 23 g) no real roots
h) ( )6.75, 3.875− and ( )2, 8− − i) ( )3.22, 1.54− − and ( )1.18, 3.98− −
j) ( )2.5, 306.25−
4. ( )0, 3.9 and ( )35, 3.72
5. a)
b) Frog: ( )21
50 25100
h d−
= − + Grasshopper: ( )21
54 3616
h d−
= − +
c) ( )40.16, 24.03
and height of 2.03 cm.
67
Solving by Substitution Assignment:
1) a) ( )6, 38− and ( )2, 6 b) 1 59
, 2 4
−
and ( )8, 19 c) ( )2, 10− and ( )2, 30
d) ( )1.52, 2.33− − and ( )1.52, 3.73 e) solutions
f) ( )2.71, 1.37− and ( )0.25, 0.78
2) 6m = , 40n =
3) ( )12, 42 and ( )13, 17−
4) 15.64t = seconds, 815.73h = meters
5) 3
2, 2
and ( )1, 3−
Solving by Elimination Assignment:
1) a) 1 9
, 2 2
b) 1
, 42
−
and ( )3, 25 c) ( )2.24, 1.94− − and ( )2.24, 15.94
d) ( )1.41, 4− and ( )1.41, 4− − e) no solution
f) ( )0.5, 6.25 and ( )0.75, 9.3125
2) 5x = , 12y =
3) 2
3r = ,
4
3C
= ,
4
9A
=
4) a) ( )3.11, 0.79− , ( )3.11, 0.79 , and ( )0, 16 b) 50 m2
Graphing Linear Inequalities in Two Variables Assignment:
1) a) ( )6, 7 and ( )12, 9 b) ( )6, 12− − , ( )4, 1− , and ( )8, 2−
c) ( )12, 4− and ( )5, 1
68
2) a)
b)
c)
d)
e)
f)
69
g)
h)
i)
3) a) 1
24
y x + b) 1
4y x
− c)
34
2y x − d)
35
4y x
− +
4) a) 12 12 250x y+ , 0x , 0y , where x represents the number of moccasins
sold and y represents the hours worked.
b)
Solving Quadratic Inequalities with One Variable Assignment:
1) a) 10 or 4, x x x x R b) 12 or 2, x x x x R − −
c) 5 7
or , 3 2
x x x x R −
d) 6 6
2 2 , 2 2
x x x R
− − +
70
e) 6 3, x x x R− f) 3 or 1, x x x x R − −
g) 3
6, 4
x x x R
h) 8 2, x x x R−
i) 2 8, x x x R j) 3 5
or , 4 3
x x x x R −
k) 1 13 1 13, x x x R− + l) 3, x x x R
m) 13 145 13 145
, 2 2
x x x R − +
n) 12 or 2, x x x x R −
2) 2 years or more
Graphing Quadratic Inequalities in Two Variables Assignment:
1) a) ( )2, 6 and ( )1, 3− b) ( )2, 2− , ( )0, 6− , and ( )2, 15− −
c) ( )4, 2− , ( )1, 3.5 , and ( )3, 2.5
2) a) 2 4 5y x x − − + b) 21
32
y x x − + c) 21
34
y x x−
− +
d) 24 5 6y x x + −
3) a)
b)
71
c)
d)
e)
f)
g)
h)
4) a) 21
12
y x + b) ( )21
1 33
y x−
− + 5) 114.6 180a
6) 20 45t lasts for 25 seconds
