Question

Which of the following points (x, y, z) is in the solution set of the system of inequalities:
x + 3y − z ≤ 5
x − 2z ≥ 0
x + y + z ≤ 10

(a) (1, 1, 3)
(b) (2, 2, 1)
(c) (6, −2, 2)
(d) (8, 3, 1)
(e) (−1, 8, −2)
(f) None of these

please explain as well :) Thank you

Answer 1

you could enter in the choices and see which satisfies the inequalities

Answer 2

so (c) is correct answer?

Answer 3

let's check
(6,-2,2)
x=6
y=-2
z=2
\[6+3(-2)-2 \le 5 \\ 6-6-2 \le 5 \\ -2 \le 5 \text{ the first inequality is true } -2 \text{ is less than } 5 \\ 6-2(2) \ge 0 \\ 6-4 \ge 0 \\ 2 \ge 0 \text{ the second inequality is true 2 is greater than 0 } \\ 6-2+2 \le 10 \\ 6 \le 10 \text{ the third inequality is true 6 is less than 10 }\]
so yes (6,-2,2) seems to check out

Answer 4

can there be more than one option?

Answer 5

I think that's the only one..

Answer 6

checking (1,1,3)
\[1+3-3 \le 5 \\1 \le 5 \text{ first inequality true }\]
\[1-2(3) \ge 0 \\ 1-6 \ge 0 \\ -5 \ge 0 \text{ \not true }\]
no need to check third inequality
(1,1,3) is not a solution

checking (2,2,1)
\[2+6-1 \le 5 \text{ is false }\]
no need to check the other inequalities
(2,2,1) is not a solution

checking (8,3,1)
\[8+9-1 \le 5 \text{ is false } \\ (8,3,1) \text{ doesn't work }\]

checking (-1,8,-2)
\[-1+3(8)+2 \le 5 \text{ is false } \\ (-1,8,-2) \text{ is \not a solution }\]

most of those choices fail at the first inequality

Answer 7

good job

Answer 8

Thank you :)