Download Solving Systems by Graphing

3.1 - Solving Systems
by Graphing
All I do is Solve!
Graph the following pairs of equations:
a. y = x + 5
y = -2x + 5
b. y = 3x + 2
y = 3x - 1
c. y = -4x - 2
y = 8x + 4
-2
System of Equations: A set of two or
more equations that use the same
variables.
Solution to a System: A set of values
that makes ALL equations true.
Is (-3, 4) a solution to the system?
Is (3,1) a solution for the following system?
ìï y = x - 2
í
ïî y = -2x + 7
Types of Solutions
1. Intersecting Lines have ONE unique solution.
1. Coincidental Lines (or same lines) have MANY
solutions.
1. Parallel Lines have NO solutions!
Solve by Graphing:
y  4 x  6

 y  2x  10
Graphing Calculator
Solve the following systems by graphing:
A.





y  x5
y  2x5
B.





y  3x2
y  3x1
C.







y  2x  2
4
y x2
2
Solving by Graphing:
 x  2y  7

2x  3y  0
Classwork (To Be Turned In):
What type of solution does each system
have? If the solution exist, what is it?
2)
1)
3)
4)
3.2 Solving Systems
Algebraically
Solving Systems by
Substitution
The Substitution Method
• Not every system can be solved easily by
graphing. Sometimes it is not always clear
from the graph where the solution is.
• We can use an algebraic method called
SUBSTITUTION to find the exact solution
without a graphing calculator.
Solving by Substitution
1. Solve for one of the variables.
2. Substitute the expression of the equation
you solved for into the other equation.
3. Solve for the variable.
4. Substitute the value of x into either
equation and solve.
Solve the system by substitution.
ìï 2x - 3y = 6
í
ïî x + y = -12
You Try! Solve the system by substitution
ìï 3x - y = 0
í
ïî 4x + 3y = 26
Solving Systems by
Elimination
Elimination
We can solve by elimination by either
Adding or Subtracting two equations to
eliminate a variable!
Solve by Elimination:
ìï 3x - 2y = 14
í
ïî 2x + 2y = 6
Solve by Elimination:
ìï 4x + 9y = 1
í
ïî 4x + 6y = -2
You Try! Solve the systems by elimination:
Note: Sometimes with elimination you will have to
multiply one or both of the equations in a system.
This creates an EQUIVALENT SYSTEM that has the
same solution to the original.
Solve the system by elimination.
Special Solutions
Solve each system by elimination.
ìï 2x - y = 3
1. í
ïî -2x + y = -3
ìï 2x - 3y = 18
2. í
ïî -2x + 3y = -6
3.6 Solving Systems
with Three Variables
3-variable Systems
Systems with 3 variables will have 3
equations. These type of systems
are in three dimensions! So it is not
going to be easy to find their
solution by graphing.
We can solve Systems with 3
variables, using Elimination OR
Substitution.
Solve by Elimination
ì x - 3y + 3z = -4
ï
í 2x + 3y - z = 15
ï 4x - 3y - z = 19
î
SO MUCH WORK!!!
Luckily, we have an easier way to do
this!
When solving system of the equations
we can use Matrices!!
Writing Systems as a Matrix Equation
For Matrix Equations in the form
AX = B
•A is called the COEFFICIENT MATRIX
•X is called the VARIABLE MATRIX
•B is called the CONSTANT MATRIX
•A Coefficient is a number INFRONT of
a variable.
•A Variable is a value represented by a
letter or symbol
•A Constant is a number WITHOUT a
variable.
Write the following System as a Matrix:
ì 2x + y + 3z = 1
ï
í 5x + y - 2z = 8
ï x - y - 9z = 5
î
Remember that when solving
matrix equations:
If AX=B then X =
-1
A B
Solve the Matrix Equation
é 2 1 3 ùé x ù é 1 ù
ê
úê
ú ê
ú
ê 5 1 -2 úê y ú = ê 8 ú
ê 1 -1 -9 úê z ú ê 5 ú
û
ë
ûë
û ë
Write as a Matrix Equations and Solve!
ì x+ y+z = 2
ï
2x + y = 5
í
ï x + 3y - 3z = 14
î
Write the follow systems as Matrix
Equations. Then Solve!
1.
2.
Unique Solutions
Remember, Systems can have 1 solution, NO
solutions, or MANY solutions.
IF matrix A’s Determinate is 0 then the matrix
does NOT have an inverse and the systems
does NOT have a unique solution.
IF matrix A’s Determine is NOT 0 then the
matrix has an inverse and the system has a
unique solutions!
Unique Solution
Determine if there is a unique solution.
ìï x + y = 3
í
ïî x - y = 7
ìï x + 2y = 5
í
ïî 2x + 4y = 8
Example
The sum of three numbers is 12. The 1st is 5 times the 2nd. The
sum of the 1st and 3rd is 9. Find the numbers.
HW 3.6/ Classwork
26.  x  2y  z  4

2x  y  4z  8
3x  y  2z  1

29.
 x  2y  2

2x  3y  z  9
 4 x  2y  5z  1

27.
 4 A  2U  I  2

5A  3U  2I  17
 A  5U  3

30. 6x  y  4z  8
 y z
  0
4 6
2x  z  2
32.  4 x  y  z  5

x  y  z  5
2x  z  1  y

28. 2l  2w  h  72

l  3w
h  2w

31.  5z  4 y
4

 3x  2y  0
 x  3z  8
