Problem to solve
Solve \[ 4x-5y=3\], \[-2x+6y=-2\]
Solution
STEP 1
Solve the following system:
{-5 y + 4 x = 3
6 y - 2 x = -2
STEP 2
Express the system in matrix form:
(4 | -5
-2 | 6)(x
y) = (3
-2)
STEP 3
Solve the system with Cramer's rule:
x = 3 | -5
-2 | 6/4 | -5
-2 | 6 and y = 4 | 3
-2 | -2/4 | -5
-2 | 6
STEP 4
Evaluate the determinant 4 | -5
-2 | 6 = 14:
x = 3 | -5
-2 | 6/14 and y = 4 | 3
-2 | -2/14
STEP 5
Evaluate the determinant 3 | -5
-2 | 6 = 8:
x = 8/14 and y = 4 | 3
-2 | -2/14
STEP 6
The gcd of 8 and 14 is 2, so 8/14 = (4×2)/(7×2) = 4/7:
x = 4/7 and y = 4 | 3
-2 | -2/14
STEP 7
Evaluate the determinant 4 | 3
-2 | -2 = -2:
x = 4/7 and y = (-2)/14
STEP 8
Divide 14 by -2: (-2)/14 = (-2)/(7×2) = -1/7:
Answer: | 
 | x = 4/7 and y = -1/7