Final answer:
To find the value of k that makes the system of equations have a solution, we need to solve the system of equations. By using the method of elimination and substitution, we find that the value of k is -21.
Step-by-step explanation:
To find the value of k that makes the system of equations have a solution, we need to solve the system of equations. The given system of equations is:
-x + z - t = 3
2x + 2y - z - 7t = 14
3x - y - 8z - 6t = k
We can solve this system of equations using the method of elimination. First, let's eliminate x from the second and third equations by multiplying the first equation by 2 and adding it to the second equation:
-2x + 2z - 2t = 6
2x + 2y - z - 7t = 14
3x - y - 8z - 6t = k
By adding these equations, we get:
2z - 2t - 7t = 20
2z - 9t = 20
Next, let's eliminate x from the third equation by multiplying the first equation by 3 and adding it to the third equation:
-3x + 3z - 3t = 9
3x - y - 8z - 6t = k
By adding these equations, we get:
3z - 11t = k + 9
Now we have a system of two equations with two variables (z and t). To solve this, we can use substitution or elimination. Let's use substitution:
From the equation 2z - 9t = 20, we can solve for z in terms of t as:
z = (20 + 9t) / 2
Substituting this value of z into the equation 3z - 11t = k + 9, we get:
3((20 + 9t) / 2) - 11t = k + 9
Simplifying this equation, we have:
30 + 27t - 11t = k + 9
16t + 30 = k + 9
16t - k = -21
Therefore, the value of k that makes the system of equations have a solution is -21.