Solve 12x 3y 3 5x 2y 1 6a 3b 2c 11 2a b 4c 15 4a 5

Solve {12x +3y = -3 5x + 2y = 1 {6a - 3b + 2c = 11 2a - b - 4c = -15 4a - 5b + 3c = 23 Solve using any method. {x - 5y + 8z = 1 2x + 9y - 4z = -8 -3x + 11y + 12z = 15 {-5x = 3y = 2 4x +2y = -1

Solution

On multiplying the 1st equation by 2, we get 8x+2y = -2…(3).Now, on subtracting the 2^nd equation from the 3^rd equation, we get 3x = -3 so that x = -3/3 = -1. On substituting x = -1 in the 1^st equation, we get 4*(-1) +y = -1 or, -4+y = -1 so that y = -1+4 = 3. The answer is x = -1,y=3.

2. 6a-3b+2c = 11…(1), 2a-b-4c = -15…(2) and 4a-5b+3c =23…(3)

On multiplying both the sides of equation 1 by 2, we get 12a-6b+4c= 22…(4). On multiplying both the sides of equation 2 by 3 and both the sides of equation 3 by 4, we get 6a-3b -12c =       -45…(5) and 16a -20b+12c = 92…(6).

Now, on adding 2^nd and 4^th equations, we get 2a-b-4c +12a-6b+4c= -15+22 or, 14a-7b = 7. Further, on dividing both the sides by 7, we get 2a-b = 1…(7)

Also, on adding 5th and 6^th equations, we get 6a-3b -12c+16a -20b+12c =-45+92 or,                     22a -23b = 47…(8)

On multiplying both the sides of the 7^th equation by 23, we get 46a -23b = 23…(9).Now, on subtracting the 8^th equation from the 9^th equation, we get 46a -23b -22a+23b = 23-47 or, 24a =          -24 so that a = -24/24 = 1. On substituting a = -1 in the7th   equation, we get 2*(-1) –b = 1 or,          -2-b = 1 so that b = -2-1 = -3. Finally, substituting a = -1 and b = -3 in the 2^nd equation, we get 2*(-1)-(-3) -4c = -15 or, 4c = -2 +3+15 or, 4c = 16 so that c = 16/4 = 4. Thus, the answer is a = -1, b= -3 and c = 4.

3. x-5y+8z= 1…(1), 2x+9y-4z= -8…(2), -3x+11y+12z = 15…(3). On multiplying both the sides of the 1^st equation by -2 and by 3, we get -2x+10y-16z= -2…(4) and 3x-15y+24z = 3…(5). Now, on adding the 2^nd and the 4^th equations, and the 3^rd and the 5^th equations , we get 2x+9y-4z-2x+10y-16z= -8-2 or, 19y -20z = -10…(6) and -3x+11y+12z +3x-15y+24z = 15+3 or, -4y +36z = 18 or ( on dividing both the sides by2), -2y +18z = 9….(7) . Now, on multiplying the 6^th equation by 2 and adding the result to 19 times the 7^th equation, we get 38y -40z -38y +342z = -20+171 or, 302z = 151. Hence z = 151/302 = ½. On substituting z = ½ in the 7^th equation, we get -2y +18*1/2 = 9 or, -2y +9 = 9 or, 2y = 9-9 = 0 so that y = 0. Finally, substituting z = 1/2 and y = 0 in the 1^st equation, we get x +8*1/2 = 1 or, x +4 = 1 so that x = 1-4 = -3. Thus, the answer is x = -3, y = 0 and z = ½.

4. -5x+3y =2…(1) and 4x+2y = -1…(2).On multiplying both the sides of the 1^st equation by 2 and the 2^nd equation by 3, we get -10x +6y = 4…(3) and 12x +6y = -3…(4). Now, on subtracting the 3^rd equation from the 4^th equation, we get 12x +6y +10x -6y = -3-4 or, 22x = -7 so that x = -7/22. On substituting x = -7/22 in the 2^nd equation, we get 4*(-7/22) +2y = -1 or, -14/11 +2y = -1 or 2y = 1+14/11 or, 2y = 3/11 so that y = 3/22. We can verify the result by substituting these values of x and y in the 1^st equation. The answer is x = -7/22 and y = 3/22.