Question

x+y+z=7 ; 2x+2y+z=4 ; -3x-2y+z=-4​

Answer 1

$\large\underline{\sf{Solution-}}$

Given equations are

$\rm :\longmapsto\:x + y + z = 7 - - - (1)$

$\rm :\longmapsto\:2x + 2y + z = 4 - - - (2)$

$\rm :\longmapsto\:3x - 2y + z = - \: 4 - - - (3)$

Now, to solve these 3 equations in 3 variables, we use method of Eliminations.

So, multiply equation (1) by 2, we get

$\rm :\longmapsto\:2x + 2y + 2z = 14 - - - (4)$

On Subtracting equation (2) from equation (4), we get

$\boxed{ \rm{ \bf :\longmapsto\:z = 10 - - - (5)}}$

On adding equation (3) and equation (2), we get

$\rm :\longmapsto\:5x + 2z = 0$

On substituting the value of z = 10, we get

$\rm :\longmapsto\:5x + 2(10) = 0$

$\rm :\longmapsto\:5x + 20 = 0$

$\rm :\longmapsto\:5x = - 20$

$\boxed{ \rm{ \bf :\longmapsto\:x = - \: 4 - - - (6)}}$

On substituting the values of x and z in equation (1), we get

$\rm :\longmapsto\: - 4 + y + 10 = 7$

$\rm :\longmapsto\: y + 6= 7$

$\rm :\longmapsto\: y= 7 - 6$

$\boxed{ \rm{ \bf :\longmapsto\:y = \:1 - - - (7)}}$

Hence,

$\red{\boxed{ \bf{ \: x = - 4, \: \: y = 1, \: \: z = 10}}}$

Verification :-

Consider, Equation (1)

$\rm :\longmapsto\:x + y + z = 7$

On substituting the values of x, y and z, we get

$\rm :\longmapsto\: - 4 + 1 + 10 = 7$

$\rm :\longmapsto\:7 = 7$

Hence, Verified

Consider, Equation (2)

$\rm :\longmapsto\:2x + 2y + z = 4$

On substituting the values of x, y and z, we get

$\rm :\longmapsto\:2( - 4)+ 2(1) + 10 = 4$

$\rm :\longmapsto\: - 8 + 2 + 10 = 4$

$\rm :\longmapsto\:4 = 4$

Hence, Verified

Consider Equation (3)

$\rm :\longmapsto\:3x - 2y + z = - 4$

On substituting the values of x, y and z, we get

$\rm :\longmapsto\:3( - 4) - 2(1) + 10 = - 4$

$\rm :\longmapsto\: - 12 - 2 + 10 = - 4$

$\rm :\longmapsto\: - 4 = - 4$

Hence, Verified