cramer’s rule gabriel cramer was a swiss mathematician (1704-1752)
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Cramer’s Rule
Gabriel Cramer was a Swiss mathematician (1704-1752)
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Coefficient Matrices You can use determinants to solve a
system of linear equations. You use the coefficient matrix of the
linear system. Linear System Coeff Matrix
ax+by=ecx+dy=f
dc
ba
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Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear System Coeff Matrix
ax+by=ecx+dy=f
If detA 0, then the system has exactly one solution:
A
df
be
xdet
and
A
fc
ea
ydet
dc
ba
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Example 1- Cramer’s Rule 2x2 Solve the system: 8x+5y=2 2x-4y=-10
42
58The coefficient matrix is:42)10()32(
42
58
and
So:
42
410
52
xand
42
102
28
y
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142
42
42
)50(8
42
410
52
x
242
84
42
480
42
102
28
y
Solution: (-1,2)
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Example 2- Cramer’s Rule 2x2
Solve the system: 2x+y=1 3x-2y=-23
The solution is: (-3,7) !!!
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Example 3- Cramer’s Rule 3x3 Solve the system: x+3y-z=1 -2x-6y+z=-3 3x+5y-2z=4 1
4
4
253
162
131
453
362
131
z
Let’s solve for Z Z=1
The answer is: (-2,0,1)!!!