Question

For the system above, which of the following are solutions?

Answer 1

10,13 and _10;0. ((. M

Answer 2

The solutions are
`(-1,5)` ,
`(-10,0)` and
`(2,8)`

Explanation:

We have a system which is formed by the following equations :

`\left \{ {{y\geq(1)/(2)x+5} \atop {2x-y<7}} \right.`

A solution to this system is a pair (x,y) that satisfies both equations.

In order to solve this exercise, we need to take each pair (x,y) and replace it in both equations. If the pair checks both equations therefore the pair (x,y) is a solution of the system.

The first pair is

`(x,y)=(-1,5)`

If we replace it in both equations :

`5\geq (1)/(2)(-1)+5` and also

`2(-1)-5<7`

⇒

`5\geq 4.5`

`-7<7`

The pair (-1,5) satisfies both equations ⇒ It is a solution for the system.

The next pair is

`(x,y)=(-4,1)`

Replacing :

`1\geq (1)/(2)(-4)+5`

`2(-4)-(1)<7`

⇒

`1\geq 3`

We find that it does not satisfy the first equation and therefore it can not be a possible solution to the system.

Now with
`(x,y)=(-10,0)`

Replacing in the equations of the system :

`0\geq (1)/(2)(-10)+5`

`2(-10)-0<7` ⇒

`0\geq 0`

`-20<7`

This pair satisfies both equations ⇒ The pair
`(x,y)=(-10,0)` is a solution of the system.

Now with
`(x,y)=(10,13)`

If we replace in the equations of the system :

`13\geq (1)/(2)(10)+5`

`2(10)-13<7` ⇒

`13\geq 10`

`7<7`

This pair satisfies the first equation but it does not satisfy the second one ⇒ It is not a solution of the system

The fifth pair is
`(x,y)=(2,8)`

Using the equations of the system :

`8\geq (1)/(2)(2)+5` and

`2(2)-8<7`

⇒
`8\geq 6`

`-4<7`

This pair verifies both equations ⇒ The pair
`(x,y)=(2,8)` is a solution of the system.

The final pair
`(x,y)=(2,-1)`

Replacing in the equations

`-1\geq (1)/(2)(2)+5`

`2(2)-(-1)<7`

⇒

`-1\geq 6` This inequality is wrong. Therefore the pair
`(x,y)=(2,-1)` it is not a solution of the system.

We conclude that the pairs
`(-1,5)` ,
`(-10,0)` and
`(2,8)` are solutions of the system above.