Download Solving 2X2 Linear Systems

M 1310
6.1
2 X 2 Linear Systems
Solving 2X2 Linear Systems
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M 1310
6.1
2 X 2 Linear Systems
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Substitution Method
Find all the solutions of the system:
2x + y = 1
3x + 4 y = 14
Solve the first equation for y:
y = 1 − 2x
Now substitute for y ( y = 1-2x) in the second equation and solve
for x:
3 x + 4(1 − 2 x ) = 14
3 x + 4 − 8 x = 14
− 5 x + 4 = 14
− 5 x = 10
x = −2
Now substitute x = –2 into either original equation:
2( − 2 ) + y = 1
−4+y =1
y=5
Check.
M 1310
6.1
2 X 2 Linear Systems
Example 1: Solve the linear system:
x+y=8
x − 3y = 0
Example 2: Solve the linear system:
3x + y = 1
x − 2 y = −9
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M 1310
6.1
2 X 2 Linear Systems
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Elimination Method
Find all solutions of the systems.
3 x + 2 y = 14
x − 2y = 2
Since the coefficients of the y term are the same except having
opposite signs. When adding the equations together the y
variable is eliminated. Most problems like these do not work out
this nice but it makes a good example:
3 x + x = 14 + 2
4 x = 16
x=4
Now substitute the 4 into either of the original equations and
solve for y:
4 − 2y = 2
− 2 y = −2
y =1
Check.
Example 3: Find all solutions:
x + 2 y = −3
2 x + 5 y = −5
M 1310
6.1
2 X 2 Linear Systems
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Example 4: Find all solutions:
5x + 2 y = 2
7x + 3y = 6
The previous examples are the problems that have one solution.
Now let’s look at problems that are examples of no solution and
infinitely many solutions.
Example 5: Find all solutions.
2x + y = 5
4x + 2y = 8
Example 6: Find all solutions.
2x + y = 4
− 6 x − 3 y = −12