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
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Unit 2 Systems of Equations & Inequalities · Unit 2 Systems of Equations & Inequalities General Outcome: • Develop algebraic and graphical reasoning through the study of relations.
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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= −