Which graph best represents the solution to the following system? −3x+y>1 −x+y≤2
The graph that best represents the solution to the given system of inequalities is the shaded region shown in the image below.
To determine which graph best represents the solution to the given system of inequalities, let's first rewrite them in slope-intercept form (y = mx + b).
1) −3x + y > 1
This inequality can be rewritten as: y > 3x + 1
2) −x + y ≤ 2
This inequality can be rewritten as: y ≤ x + 2
To plot these lines on a graph:
1) Start with the equation y > 3x + 1:
- Plot the y-intercept: (0, 1)
- Determine additional points by using the slope, which is 3. For every unit increased in x, y increases by 3, so plot another point at (1, 4).
- Draw a dashed line through these points, indicating that the line is not included in the solution.
2) Next, plot the equation y ≤ x + 2:
- Plot the y-intercept: (0, 2)
- Determine additional points using the slope, which is 1. For every unit increased in x, y increases by 1, so plot another point at (1, 3).
- Draw a solid line through these points, indicating that the line is included in the solution.
Finally, shade the appropriate region based on the inequalities:
- Since the inequality for the first line is y > 3x + 1, shade the region above the dashed line.
- Since the inequality for the second line is y ≤ x + 2, shade the region below the solid line.
The graph that best represents the solution to the system of inequalities is the shaded region where the shaded regions overlap, indicating the common solution area.
To graphically represent a system of inequalities, we typically start by graphing each inequality on the same coordinate plane. Then, we can find the intersection of the shaded regions to determine the solution to the system.
Let's start by graphing the inequality −3x + y > 1:
Step 1: Begin by graphing the equation −3x + y = 1, which is the boundary line for this inequality. To plot this line, we need two points. You can choose any values for x and y and then solve the equation to find the corresponding points.
Let's choose x = 0:
-3(0) + y = 1
y = 1
So, one point is (0, 1).
Now, let's choose y = 0:
-3x + 0 = 1
-3x = 1
x = -1/3
Another point is (-1/3, 0).
Step 2: Plot the two points on the coordinate plane and draw a line passing through them. This line represents the equation −3x + y = 1.
Step 3: Since the inequality is ">," the shaded region will be above the line. You can shade the region above the line, excluding the line itself.
Now, let's move on to the next inequality: −x + y ≤ 2.
Step 4: Let's graph the equation −x + y = 2, which is the boundary line for this inequality. Using a similar approach as before, we can find two points:
For x = 0:
-(0) + y = 2
y = 2
One point is (0, 2).
For y = 0:
-(x) + 0 = 2
x = -2
Another point is (-2, 0).
Step 5: Plot the two points on the coordinate plane and draw a line passing through them. This line represents the equation −x + y = 2.
Step 6: Since the inequality is "≤," the shaded region will be below the line or on the line itself. You can shade the region below the line, including the line.
Finally, we need to find the intersection of the shaded regions, which represents the solution to the system of inequalities. In this case, the shaded region above the line −3x + y = 1 and below the line −x + y = 2 intersect in the region that satisfies both inequalities. That area represents the solution to the system.
The graph of the system of inequalities is the shaded region that is above the line −3x + y = 1 and below the line −x + y = 2.