Question

For what values of k the set of equations 2x – 3y + 6z -5t = 3, y – 4z + t = 1, 4x – 5y + 8z – 9t = k has infinites number of solutions.

Answer 1

Hint: Observe the given three system of equations and try to eliminate x, y, z and t to get value of ‘k’. subtract equation two and three given in the problem and relate it with the first equation and eliminate x, y, z by applying some operations with both equations. Get ‘t’ after elimination.

Complete step-by-step answer:

Given system of equation in the problem are
2x – 3y + 6z – 5t = 3 …………… (i)
y – 4z + t = 1 ………………….(ii)
4x – 5y + 8z – 9t = k ……………… (iii)
As the given system of equations has an infinite number of solutions. It means we cannot get exact values of x or y or z, we will get some relation among them. So, we do not need to calculate values of x, y or z.
Let us try to eliminate x, y, z from the given equations to get the value of k.
So, let us subtract equation (ii) and (iii). So, we get
(4x – 5y + 8z – 9t = k) – (y – 4z + t = 1) = k – 1
4x – 5y + 8z – 9t - y + 4z – t = k - 1
4x – 6y + 12z – 10t = k – 1………. (iv)
Now, we can observe the equation (i) and (iv) and hence, get that the coefficients of x, y, z, t of equation (iv) are twice of the coefficients of x, y, z, t of equations (i). so, divide the equation (iv) by 2, hence, we get
$\dfrac{4x-6y+12z-10t}{2}=\dfrac{k-1}{2}$
$2x-3y+6z-5t=\dfrac{k-1}{2}$ …….. (v)
Now, we can observe equation (i) and (v) replace the left hand side of the above equation by ‘3’, as the value of 2x – 3y + 6z – 5t is 3 from equation (i). so, we get
$\dfrac{3}{1}=\dfrac{k-1}{2}$
One cross-multiplying the above equation, we get
6 = k – 1
k = 7
Hence, if the system of given questions has infinite solutions, then the value of k is 7.

Note: One may try to get values of x, y, z but it will not give or answer. And one may apply $\left| A \right|=0$, where A is the determinant formed by the coefficients of x, y, z, but we will not get the value of k from here as well. So, try to eliminate x, y, z and t to get the required answer. And this can be done easily only by observing the coefficients of the equation.
So, observation is the key point of this question.
Involvement of five variables make this question complex, but take x, y, z variables only and try to solve the question. Don’t confuse it with the number of variables.