Fill in the Blank Sketch the region enclosed by the graphs of x = 0, 6y – 5x = 0, and x + 3y = 21. Find the area.
To sketch the region enclosed by the graphs, we first need to find the points of intersection between the lines.
1. x = 0:
This is the y-axis, so one of the boundary lines.
2. 6y - 5x = 0:
Rearranging this equation to solve for y, we get y = 5x/6.
Now, plug in x = 0 from the first equation to get y = 0. So, this line intersects the x-axis at x = 0 and y = 0.
3. x + 3y = 21:
Now, Rearranging this equation to solve for y, we get y = (21 - x)/3.
Now plug in x = 0 to get y = 7. So this line intersects the x-axis at x = 0 and y = 7.
Now we have the following points of intersection: (0,0), (0,7), (0,0).
Now, find the area of the region enclosed by the lines. Since we have two vertical lines (x=0 and x=6) and one slanted line (from the second equation), we can see that the shape is a rectangle with a slanted top.
The width of the rectangle is 6 (from x=0 to x=6), and the height is 7 (from y=0 to y=7).
So, the area of the region enclosed by the graphs is 6 * 7 = 42 square units.
Fill in the Blank Sketch the region enclosed by the graphs of x = 0, 6y – 5x = 0, and x + 3y = 21. Find the area.
To sketch the region enclosed by the graphs, we first need to find the points of intersection between the lines.
1. x = 0:
This is the y-axis, so one of the boundary lines.
2. 6y - 5x = 0:
Rearranging this equation to solve for y, we get y = 5x/6 or y = (5/6)x.
Now let's find the point of intersection with x = 0:
y = (5/6)(0)
y = 0
So the line intersects the y-axis at (0,0).
3. x + 3y = 21:
Rearranging this equation to solve for y, we get y = (21 - x)/3.
Let's find the point of intersection with x = 0:
y = (21 - 0) / 3
y = 7
So this line intersects the y-axis at (0,7).
Now we have the following points of intersection: (0,0), (0,7).
Now we can see that the region enclosed by the graphs is a triangle with base 7 (from the y-axis to the line x + 3y = 21) and height 6 (from the x-axis to the line 6y - 5x = 0).
Therefore, the area of the triangle is (1/2) * base * height = (1/2) * 7 * 6 = 21 square units.