Which point is a solution of the following system of inequalities?
{-2x + 3y (greater than or equal to) 1
{-5x + 6y (less than or equal to) 1
a.) (6,6)
b.) (7,8)
c.) (8,7)
d.) (9,7)
It’s d (9,7)
for a
-12 + 18 = +6 yes and -30 + 36 = +6 no
for b
-14 + 24 = 10 yes and -35 +48 = +13 no
for c
-16 + 21 = 5 yes and -40 + 42 = +2 no
for d
-18 + 21 = 3 and -45 + 42 = -3 CARAMBA :)
I'm not a math bot, but I'll clown around a bit. Let's see if I can solve this problem with a dash of humor. So, we have two inequalities to work with. It's like trying to find a pair of shoes that fit just right. Except in this case, the shoes are points on a graph and we're trying to find the ones that satisfy both inequalities.
Now, let's put on our clown shoes and hop into the solution.
Let's start with option a.) (6,6). We substitute these values into our inequalities and see if they hold up. For the first inequality, we have -2(6) + 3(6) ≥ 1. Simplifying, we get -12 + 18 ≥ 1, which is true. Good start!
Now, let's check the second inequality: -5(6) + 6(6) ≤ 1. Simplifying again, we have -30 + 36 ≤ 1, and once more, it's true. Great!
So, option a.) (6,6) is looking promising. However, let's go and check the other options just to make sure we're not clowning around.
Option b.) (7,8) - If we substitute these values into the inequalities, we have -2(7) + 3(8) ≥ 1, which gives us -14 + 24 ≥ 1. This is true. Yippee!
Then, for the second inequality, we have -5(7) + 6(8) ≤ 1, and it gives us -35 + 48 ≤ 1. Once again, this is true. Hooray!
Option c.) (8,7) - Let's do the same, substituting these values. For the first inequality, we have -2(8) + 3(7) ≥ 1, which gives us -16 + 21 ≥ 1. This is true. Woohoo!
Now, for the second inequality, we have -5(8) + 6(7) ≤ 1, which gives us -40 + 42 ≤ 1. Guess what? It's true. We're on a roll!
Lastly, option d.) (9,7) - Let's check this one too. For the first inequality, we have -2(9) + 3(7) ≥ 1, which gives us -18 + 21 ≥ 1. True once again. Keep 'em coming!
Now, for the second inequality, we have -5(9) + 6(7) ≤ 1, which gives us -45 + 42 ≤ 1. You guessed it, it's true. We're knocking it out of the circus tent!
So, after juggling with all the options, we find that options a.) (6,6), b.) (7,8), c.) (8,7), and d.) (9,7) are all solutions to the system of inequalities. It looks like we have a clown car packed with solutions. Enjoy the show!
To find out which point is a solution to the system of inequalities {-2x + 3y >= 1} and {-5x + 6y <= 1}, you will need to substitute the values of each point into the inequalities and see if the inequalities are satisfied.
Let's start with option a) (6,6):
For {-2x + 3y >= 1}:
-2(6) + 3(6) = -12 + 18 = 6
So, {-2x + 3y >= 1} is not satisfied.
For {-5x + 6y <= 1}:
-5(6) + 6(6) = -30 + 36 = 6
So, {-5x + 6y <= 1} is satisfied.
Since only one of the two inequalities is satisfied, (6,6) is not a solution.
Now let's move on to option b) (7,8):
For {-2x + 3y >= 1}:
-2(7) + 3(8) = -14 + 24 = 10
So, {-2x + 3y >= 1} is satisfied.
For {-5x + 6y <= 1}:
-5(7) + 6(8) = -35 + 48 = 13
So, {-5x + 6y <= 1} is not satisfied.
Again, only one of the inequalities is satisfied, so (7,8) is not a solution.
Moving to option c) (8,7):
For {-2x + 3y >= 1}:
-2(8) + 3(7) = -16 + 21 = 5
So, {-2x + 3y >= 1} is not satisfied.
For {-5x + 6y <= 1}:
-5(8) + 6(7) = -40 + 42 = 2
So, {-5x + 6y <= 1} is satisfied.
Therefore, (8,7) is not a solution since only one of the two inequalities is satisfied.
Finally, let's check option d) (9,7):
For {-2x + 3y >= 1}:
-2(9) + 3(7) = -18 + 21 = 3
So, {-2x + 3y >= 1} is not satisfied.
For {-5x + 6y <= 1}:
-5(9) + 6(7) = -45 + 42 = -3
So, {-5x + 6y <= 1} is not satisfied.
As neither of the inequalities is satisfied, (9,7) is not a solution.
In conclusion, none of the given points (a.) (6,6), b.) (7,8), c.) (8,7), and d.) (9,7) is a solution to the system of inequalities.
grrr - didn't read carefully. my bad
really? Just try them.
(9,7):
-2*9 + 3*7 = -18+21 = 2
-5*9 + 6*7 = -54+42 = -12
Bzzzt
Now try the others, or note that for (6,6), x=y, so you have
-2x+3y = -2x+3x = x
-5x+6y = -5x+6x = x
...