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3.1 Intro to Systems
Contributed by: Smith
(Original author: Pappal)
You are solving for the variable that makes the
equation true (equal).
When you solve one variable equations
2x - 4 = 6
You are solving for the variable that makes the
equation true (equal).
When you solve one variable equations
2x - 4 = 6
+4
+4
Add 4 to both sides
You are solving for the variable that makes the
equation true (equal).
When you solve one variable equations
2x - 4 = 6
2x = 10
+4
+4
Fill in the correct value
You are solving for the variable that makes the
equation true (equal).
When you solve one variable equations
2x - 4 = 6
2x = 10
+4
2
+4
2
You are solving for the variable that makes the
equation true (equal).
When you solve one variable equations
2x - 4 = 6
2x = 10
+4
2
x = 5
+4
2
You are solving for the variable that makes the
equation true (equal).
When you solve one variable equations
2x - 4 = 6
2x = 10
+4
2
x = 5
+4
2
You are solving for the number that can go in for x
When you solve one variable equations
2x - 4 = 6
2x - 4 = 6
2x = 10
2
x = 5
x = 5
+4
2
+4
2( 5 ) -4 = 6
Plug in 5 for x
in the original
equation to check
There are many sets of possible solutions
For a two variable equations
x + y = 5
such as:
x = 1
There are many sets of possible solutions
y = 4
For a two variable equations
For a two variable equations
because 1 + 4 = 5
x + y = 5
such as:
x =2
x = 1
Fill in the correct value
There are many sets of possible solutions

y = 3

y = 4
For a two variable equations
For a two variable equations
because 2 + 3 = 5
because 1 + 4 = 5
x + y = 5
such as:
x =2
x = 100
x = 1
There are many sets of possible solutions

y = 3

y = 4
y = -95
For a two variable equations
For a two variable equations
because 2 + 3 = 5
because 1 + 4 = 5
because 100 - 95 = 5
x + y = 5
Fill in the correct value
such as:
x =2
x = 100
x = 1
In fact, the number of solutions are infinite
There are many sets of possible solutions

y = 3

y = 4
y = -95
For a two variable equations
For a two variable equations
because 2 + 3 = 5
because 1 + 4 = 5
because 100 - 95 = 5
x + y = 5
such as:
But if I have two equations
each with two variables
x + y = 5
x - y = -3
I can solve for the x and y values that work for BOTH
The x and y values that make both equations true
But if I have two equations
each with two variables
x + y = 5
x - y = -3
I can solve for the x and y values that work for BOTH
Solutions
Possible
But if I have two equations
each with two variables
x + y = 5
x - y = -3
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
But if I have two equations
each with two variables
x + y = 5
x - y = -3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Fill in the correct value
Possible
But if I have two equations
each with two variables
x + y = 5
x - y = -3
x = 2  y = 3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
Fill in the correct value
But if I have two equations
each with two variables
x + y = 5
x - y = -3
x = 2  y = 3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
But if I have two equations
each with two variables
x + y = 5
x - y = -3
Solutions
Possible
x = 2  y = 3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
But if I have two equations
each with two variables
x + y = 5
x - y = -3
x = 0  y = 3
Solutions
Possible
x = 2  y = 3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
But if I have two equations
each with two variables
x + y = 5
x - y = -3
x = 0  y = 3
x = 1  y = 4
Solutions
Possible
x = 2  y = 3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
But if I have two equations
Drag in the values
each with two variables
x + y = 5
x - y = -3
x =     y =  
x = 0  y = 3
x = 1  y = 4
Solutions
Possible
2
?
5
?
x = 2  y = 3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
But if I have two equations
The set that matches
each with two variables
is the answer
x + y = 5
x - y = -3
x = 2  y = 5
x = 0  y = 3
x = 1  y = 4
Solutions
Possible
x = 2  y = 3
x = 1  y = 4
I can solve for the x and y values that work for BOTH
x = 0  y = 5
Solutions
Possible
But if I have two equations
The set that MATCHES
each with two variables
is the answer
x + y = 5
x - y = -3
x = 2  y = 5
x = 0  y = 3
x = 1  y = 4
Solutions
Possible
When x=1 and y=4 which equation(s) are true (equal)?
2nd equation
1st equation
x = 1 and y = 4 is the solution to this system
1st equation only
2nd equation only
Neither equation
 Both equations
x + y = 5
x - y = -3
Substitute x=4, y=1 (4,1) into both equations
to see if (4,1) is a solution of the system of equations.
2nd equation
1st equation
When x=4 and y=1
which equation(s)
are true?
2x - y =  4
x + y = 5
1st equation only
2nd equation only
Neither equation
 Both equations
Substitute x=4, y=1 (4,1) into both equations
to see if (4,1) is a solution of the system of equations.
2nd equation
1st equation
When x=4 and y=1
which equation(s)
are true?
2x - y =  4
x + y = 5
Point (4,1) only
Sat-is-fies
the 1st equation.

(4,1) is NOT a solution
to the entire system
Substitute x=2, y=0 (2,0) into both equations
to see if (2,0) is a solution of the system of equations.
2nd equation
1st equation
When x=2 and y=0
which equation(s)
are true?
2x - y =  4
x + y = 5
1st equation only
2nd equation only
Neither equation
 Both equations
Substitute x=2, y=0 (2,0) into both equations
to see if (2,0) is a solution of the system of equations.
2nd equation
1st equation
When x=2 and y=0
which equation(s)
are true?
2x - y =  4
x + y = 5
Point (2,0) only
Sat-is-fies
the 2nd equation.

(2,0) is NOT a solution
to the entire system
Substitute x=3, y=2 (3,2) into both equations
to see if (3,2) is a solution of the system of equations.
2nd equation
1st equation
When x=3 and y=2
which equation(s)
are true?
2x - y =  4
x + y = 5
1st equation only
2nd equation only
Neither equation
 Both equations
Substitute x=3, y=2 (3,2) into both equations
to see if (3,2) is a solution of the system of equations.
2nd equation
1st equation
When x=3 and y=2
which equation(s)
are true?
2x - y =  4
x + y = 5
Both equations are true.
(3,2) is a solution of this
system of equations
The End

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