Which point is a solution to the following system of inequalities?
-2x+3y>1
-5x+6y<1
To find the solution to the system of inequalities, we need to find a point that satisfies both inequalities simultaneously.
Let's solve each inequality separately:
Inequality 1: -2x + 3y > 1
We can rewrite this inequality as 3y > 2x + 1.
To find a point that satisfies this inequality, we can choose any value of x and solve for y.
For example, let's choose x = 0:
3y > 2(0) + 1
3y > 1
y > 1/3
So, any point where y is greater than 1/3 satisfies the first inequality.
Inequality 2: -5x + 6y < 1
We can rewrite this inequality as 6y < 5x + 1.
To find a point that satisfies this inequality, we can choose any value of x and solve for y.
For example, let's choose x = 0:
6y < 5(0) + 1
6y < 1
y < 1/6
So, any point where y is less than 1/6 satisfies the second inequality.
Now, let's find a point that satisfies both inequalities simultaneously:
Let's choose y = 1/4, which satisfies both y > 1/3 and y < 1/6.
Plug this value into one of the inequalities to find x.
Let's use the first inequality:
-2x + 3(1/4) > 1
-2x + 3/4 > 1
-2x > 1 - 3/4
-2x > 1/4
x < -1/8
So, the point (-1/8, 1/4) is a solution to the system of inequalities.