Write a system of inequalities to describe the shaded area of the graph.
This is the question on my homework today that i just don't get. My teacher said to use point slope form.
y=y1+m(x-x1)
The coordianate points of the triangular area are
line 1: (2,1),(7,9)
line 2: (7,9),(11,3)
line 3: (2,1), (11,3)
(i wish that i could just show you the graph.
Kay, so i know that lines 1 and 2 are going to be less than or equal to, and line 3 will be greater than or equal to. However, after I put it into the formula, (which i don't think i got quite right) i don't know if that's the final part of writing the system.
for line 1, i think that i got something like:
(7,9) y=9+8/5(x-7)
Is this right? Also, is this the last part of making this fit into a system?(i think it looks too wierd to leave like that) Any help or tips is VERY much appreciated. Thanks.
I assume the acceptable values are in the triangle.
so for the bottom line:
y=mx+b the two points are 2,1 and 11,3
y=2/9 x+ b ( m= (y1-y2)/(x1-x2)
then to solve for b
3=2/9 *11 + b
b=3-22/9=5/9
so the solution area here, is all above the line, or y>2x/9 + 5/9
Now consider the left line
y=mx+ b (m= (9-1)/7-2)=8/5
y=8x/5+b solve for b...
9=8*7/5+b
b=9-56/5=-11/5
and the solution set is below the line,or
y<8x/5-11/5
Then to the last line the same way, the solution set will be below that line.
Those three equations are your system of inequalities.
To describe the shaded area of the graph using a system of inequalities, you need to graph each line and identify the region where the lines overlap.
Let's start with line 1:
Using the point-slope form, the equation for line 1 can be written as:
y - 1 = (9 - 1) / (7 - 2) * (x - 2)
Simplifying:
y - 1 = 8/5 * (x - 2)
y - 1 = 8/5x - 16/5
y = 8/5x - 16/5 + 1
y = 8/5x - 16/5 + 5/5
y = 8/5x - 11/5
For line 2:
Using the point-slope form, the equation for line 2 can be written as:
y - 9 = (3 - 9) / (11 - 7) * (x - 7)
Simplifying:
y - 9 = -6/4 * (x - 7)
y - 9 = -3/2x + 21/2
y = -3/2x + 21/2 + 9
y = -3/2x + 21/2 + 18/2
y = -3/2x + 39/2
For line 3:
Using the point-slope form, the equation for line 3 can be written as:
y - 1 = (3 - 1) / (11 - 2) * (x - 2)
Simplifying:
y - 1 = 2/9 * (x - 2)
y - 1 = 2/9x - 4/9
y = 2/9x - 4/9 + 1
y = 2/9x - 4/9 + 9/9
y = 2/9x + 5/9
Now we have the equations for all three lines. To describe the shaded area, we need to set up inequalities.
We observe that the shaded region is bounded by lines 1, 2, and 3. Therefore, the system of inequalities to describe the shaded area is:
1) y ≤ 8/5x - 11/5 (equation for line 1, representing the region below line 1)
2) y ≤ -3/2x + 39/2 (equation for line 2, representing the region below line 2)
3) y ≥ 2/9x + 5/9 (equation for line 3, representing the region above line 3)
Solving this system of inequalities will give you the region that describes the shaded area on the graph.
To describe the shaded area of the graph using a system of inequalities, we need to find the equations of the three lines that form the boundary of the shaded region.
Let's start with Line 1: (2,1) and (7,9).
Using the point-slope form equation that you mentioned, we have:
y - y1 = m(x - x1)
Substituting the given points (2,1) and (7,9), we get:
y - 1 = m(x - 2)
To find the value of 'm,' we need the slope. Using the slope formula:
m = (y2 - y1) / (x2 - x1)
m = (9 - 1) / (7 - 2)
m = 8/5
Now, substitute the value of 'm' into the equation:
y - 1 = (8/5)(x - 2)
y - 1 = 8/5x - 16/5
y = 8/5x - 16/5 + 1
y = 8/5x - 16/5 + 5/5
y = 8/5x - 11/5
So, the equation for Line 1 is: y = 8/5x - 11/5
Repeat the same process for Lines 2 and 3.
Line 2: (7,9) and (11,3)
Find the slope:
m = (3 - 9) / (11 - 7)
m = -6/4
m = -3/2
Using point-slope form:
y - 9 = -3/2(x - 7)
y - 9 = -3/2x + 21/2
y = -3/2x + 21/2 + 18/2
y = -3/2x + 39/2
So, the equation for Line 2 is: y = -3/2x + 39/2
Line 3: (2,1) and (11,3)
Find the slope:
m = (3 - 1) / (11 - 2)
m = 2/9
Using point-slope form:
y - 1 = 2/9(x - 2)
y - 1 = 2/9x - 4/9
y = 2/9x - 4/9 + 9/9
y = 2/9x + 5/9
So, the equation for Line 3 is: y = 2/9x + 5/9
Now that we have the equations for all three lines, we can write the system of inequalities to describe the shaded area:
y ≤ 8/5x - 11/5
y ≤ -3/2x + 39/2
y ≥ 2/9x + 5/9
These inequalities represent the boundaries of the shaded area on the graph.