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UNIT 6
Solving Systems of Equations by Graphing
You can solve a system of equations (2 or more equations) by graphing.
Graph all equations on the same graph. The solution is the intersection point/s of the
graphs.
Graph the following system of equations (by hand) and state the solution/s if they
exist.
1. y  3 x , and y  x  2
2. 3 x  y  2 , and y  3 x  5
(hint: put equations in slope-int. form)
y
y
x
x
Solution/s: ____________________
Solution/s: ____________________
3. 4 x  2 y  8 , and 12 x  6 y  24
4. y  ( x  3) 2  1 , and y  3
y
y
x
x
Solution/s: ____________________
Solution/s: ____________________
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5. y  2 x , and y   x
y
x
Solution/s: ____________________
A graphing calculator can be used to find an approximate solution to a system of
equations (the solution may be a decimal which has been rounded).
Calculator Steps:
1. Enter equations in y1=, y2=, etc. and find an appropriate window where you can
see all intersection points.
2. “calc”, “intersection”
3. Move curser close to the solution you are finding then answer: First curve?, enter,
Second Curve?, enter, Guess?, enter
4. Repeat process if system contains additional solutions.
Solve the following systems by graphing (calculator).
6. y 
1
x  5 , and y  ( x  2) 2  3
2
7. y  1.8 x 2  19 x  40 , and  5.4 x  y  14
Solution/s: ____________________
Solution/s: ____________________
8. (Page 363) The United States is researching a project called the National Aerospace
Plane, which would regularly fly passengers into space. A model rocket launch was tested
as part of this research project.
1
The model rocket flight can be modeled by the equation: h(t )  88.2t  (9.8)t 2
2
(where h is height in meters and t is time in seconds)
How many seconds after take-off did the rocket reach a height of 200 feet?
Hint: Graph the “model” equation and graph a line at 200 feet.
Solution/s: ____________________
Assignment #12
worksheet Solving Systems by Graphing
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3.2: Solving Systems of Equations Algebraically
There are two different algebraic methods for solving a system of two equations.
Substitution Method:
Steps:
1. Solve one equation for a variable (a variable with a coefficient of 1 is easiest if this is
possible).
2. Substitute for this variable the other equation.
3. You should only have one variable in this new equation. Solve for a value/s.
4. Use substitution to find the remaining part of the ordered pair/s.
Solve the following systems using substitution. The systems may have 0, 1, 2, or
infinite solutions.
1.
1
 x y 7
2
4 x  2 y  16
8 x  4 y  5
2. 
 y  2x  4
Solution/s: ____________________
Solution/s: ____________________
 y  x2
3. 
y  x  6
 y  x  3
4. 
 y  4
Solution/s: ____________________
Solution/s: ____________________
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Addition / Elimination Method
Steps.
1. Algebraically put equations in the same “form”.
2. If necessary, multiply one or both equations by a constant so that when the equations
are added together vertically one variable will “eliminate”.
3. Solve for the value of the remaining variable.
4. Use substitution to find the remaining part of the ordered pair for all solutions.
Solve the following systems using Addition / Elimination. The systems may have 0, 1,
2, or infinite solutions.
3x  y  7
1. 
4 x  2 y  16
5 x  4 y  12
2. 
6 y  7 x  40
Solution/s: ____________________
Solution/s: ____________________
 y  x 2  4
3. 
2
 y   x  6
5m  2n  8
4. 
4m  3n  2
Solution/s: ____________________
Solution/s: ____________________
Assignment #13
worksheet
Solving Systems of Equations Algebraically
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3.3, 4.9: Solving Systems of Inequalities by Graphing
The intersection of the shaded regions is the solution of the system of inequalities. If the
regions to not intersect, then no real solutions exist.
Solve each system of inequalities by graphing. (The graph is a picture of the solution
set.)
y  x  2
1. 
x  2 y  1
y  3
2. 
x  2
y
y
x
x
 y  2( x  3) 2

3. 
1
y  x  5
3

 y   x 2  4
4. 
 y  x  1
y
y
x
x
Assignment #14 Complete the square where necessary:
page 171 # 1, 3, 4 – 8, 11, 14, 15, 16, 44, 45; Page 304 # 20, 21, 23, 24, 79, 80, 83
Assignment #15
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Solving Systems of Equations with Three Variables
Systems of three equations can be solved with substitution or addition method. Graphically
the solution point is a 3-dimensional point (x, y, z)
Solve each system of equations.
1. Substitution Method
2. Substitution Method
x  2 y  z  9

3 y  z  1
3z  12

x  2 y  z  4

4 y  3z  1
 y  5z  6

3. Addition / Elimination Method
2 x  y  2 z  15

 x  y  z  3
3x  y  2 z  18

Assignment #16
Assignment #17
worksheet
worksheet
4. Addition / Elimination Method
2 x  y  z  3

 x  3 y  2 z  11
3 x  2 y  4 z  1

Unit 6 Review
Unit 6 Test (with review) is next class – Be Prepared!
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