Which point is a solution to the following system of inequalities?
-2x+3y greater than or equal to 1
-5x+6y less that or equal to 1
There are infinite solutions to the system of inequalities, as it represents a region rather than a single point.
To find the solution to the given system of inequalities, we need to graph the inequalities on a graph and find the intersection point.
First, let's graph the first inequality: -2x + 3y ≥ 1.
To graph this inequality, we can start by graphing the corresponding equation: -2x + 3y = 1.
Let's rearrange it to solve for y:
3y = 2x + 1
y = (2/3)x + 1/3.
Now, let's plot this line on the graph:
- Start by plotting the y-intercept of 1/3 which is the point (0, 1/3).
- Then, use the slope of 2/3 to find two more points.
- Move up 2 units and right 3 units from the y-intercept to get (3, 7/3).
- Move down 2 units and left 3 units from the y-intercept to get (-3, -5/3).
Now, draw the line passing through these three points.
Next, let's graph the second inequality: -5x + 6y ≤ 1.
To graph this inequality, we can start by graphing the corresponding equation: -5x + 6y = 1.
Let's rearrange it to solve for y:
6y = 5x + 1
y = (5/6)x + 1/6.
Now, let's plot this line on the graph:
- Start by plotting the y-intercept of 1/6 which is the point (0, 1/6).
- Then, use the slope of 5/6 to find two more points.
- Move up 5 units and right 6 units from the y-intercept to get (6, 11/6).
- Move down 5 units and left 6 units from the y-intercept to get (-6, -9/6).
Now, draw the line passing through these three points.
The solution to the system of inequalities is the region where the two lines intersect. In this case, there is exactly one intersection point.
By observing the graph, the point of intersection is approximately (-2, 4/3), which means (-2, 4/3) is the solution to the given system of inequalities.
To find the point that is a solution to the system of inequalities, we need to solve both inequalities simultaneously.
Let's start with the first inequality: -2x + 3y ≥ 1.
Step 1: Solve for y:
-2x + 3y ≥ 1
3y ≥ 2x + 1
y ≥ (2/3)x + 1/3
This inequality represents a region above the line y = (2/3)x + 1/3, including the line itself.
Step 2: Graph the region:
Plot the line y = (2/3)x + 1/3 (which has a slope of 2/3 and a y-intercept of 1/3) and shade the region above the line.
Now let's move to the second inequality: -5x + 6y ≤ 1.
Step 1: Solve for y:
-5x + 6y ≤ 1
6y ≤ 5x + 1
y ≤ (5/6)x + 1/6
This inequality represents a region below the line y = (5/6)x + 1/6, including the line itself.
Step 2: Graph the region:
Plot the line y = (5/6)x + 1/6 (which has a slope of 5/6 and a y-intercept of 1/6) and shade the region below the line.
The solution to the system of inequalities is the point where both shaded regions overlap.
Finally, to find the specific point, you can either analyze the graph visually or find the point(s) of intersection using algebraic methods.
If you need more information, please provide the specific domain or range of values for x and y.