Consider the following system of equations.2x + 3y + 4z = 135x + 7y+7z = 269x + 13y + 15z = 13λthen the value ofλ for which the system has infinetly many solution is ___________Correct answer is '4'. Can you explain this answer?

Question

Consider the following system of equations.2x + 3y + 4z = 135x + 7y+7z = 269x + 13y + 15z = 13&lambda;then the value of&lambda; for which the system has infinetly many solution is ___________Correct answer is '4'. Can you explain this answer?

Answer

To solve this system of equations, we can use the method of elimination or substitution. Let's use the method of substitution.

From the first equation, we can solve for x in terms of y and z:
2x = 13 - 3y - 4z
x = (13 - 3y - 4z)/2

Now, we substitute this expression for x into the second equation:
(13 - 3y - 4z)/2 + 7y + 7z = 26

Next, we simplify and solve for y:
13 - 3y - 4z + 14y + 14z = 52
11y + 10z = 39
11y = 39 - 10z
y = (39 - 10z)/11

Now, we substitute the expressions for x and y into the third equation:
(13 - 3(39 - 10z)/11 - 4z)/2 + 13(39 - 10z)/11 + 15z = 13

Next, we simplify and solve for z:
(13 - (117 - 30z)/11 - 4z)/2 + (507 - 130z)/11 + 15z = 13
(13 - (117 - 30z) - 44z)/2 + (507 - 130z) + 165z/11 = 13
(13 - 117 + 30z - 44z)/2 + 507 - 130z + 165z/11 = 13
(-104 - 14z)/2 + 507 - 130z + 165z/11 = 13
-52 - 7z + 507 - 130z + 15z = 26
-117z = -31
z = -31/(-117)
z = 31/117
z = 1/3

Now, we substitute this value of z back into the expression for y:
y = (39 - 10(1/3))/11
y = (39 - 10/3)/11
y = (117 - 10)/33
y = 107/33

Finally, substitute the values of y and z back into the expression for x:
x = (13 - 3(107/33) - 4(1/3))/2
x = (13 - 321/33 - 4/3)/2
x = (429 - 321 - 44)/66
x = 64/11

Therefore, the solution to the system of equations is x = 64/11, y = 107/33, and z = 1/3.

Answer

λ= 4

The a ugm ented matrix is given by ,

for infinitely many solutions ,

Rank (A) = Rank [ A , b ] = 2

So 1 3λ - 5 2 = 0 => λ =4

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