y ≤ = -1/3x + 7
y ≥ = -x + 3
solve the system of inequalities
To solve this system of inequalities, we need to find the values of x and y that satisfy both inequalities.
First, let's graph the first inequality y ≤ -1/3x + 7.
Start by graphing the equation y = -1/3x + 7 as a solid line.
The slope of the line is -1/3, which means that for every 1 unit increase in x, y decreases by 1/3 units.
The y-intercept is 7, so the line passes through the point (0, 7).
To find another point on the line, we can use the slope. If we increase x by 3 units, y should decrease by 1 unit. So we can use (3, 7 - 1/3 * 3) = (3, 6) as another point.
Plot these two points and draw a line through them.
Now, since we have y ≤ the equation, we need to shade the area below the line.
Next, let's graph the second inequality y ≥ -x + 3.
Start by graphing the equation y = -x + 3 as a solid line.
The slope of the line is -1, which means that for every 1 unit increase in x, y decreases by 1 unit.
The y-intercept is 3, so the line passes through the point (0, 3).
To find another point on the line, we can use the slope. If we increase x by 1 unit, y should decrease by 1 unit. So we can use (1, 3 - 1 * 1) = (1, 2) as another point.
Plot these two points and draw a line through them.
Now, since we have y ≥ the equation, we need to shade the area above the line.
The solution to the system of inequalities is the overlapping shaded region.
In this case, the overlapping shaded region is the area where the shaded region below the first line and the shaded region above the second line intersect.
This can be seen as the triangular region between the two lines.
To express this solution region algebraically, we can write it as:
-1/3x + 7 ≤ y ≤ -x + 3
where do i shade?
To shade the solution region, you shade the area where both inequalities are true at the same time. In this case, you need to shade the region that is below the line y ≤ -1/3x + 7 and above the line y ≥ -x + 3.
To shade the region below the line y ≤ -1/3x + 7, you would shade the area below the line, including the line itself. This can be represented by shading the region below the line in your graph.
To shade the region above the line y ≥ -x + 3, you would shade the area above the line, including the line itself. This can be represented by shading the region above the line in your graph.
The region where both inequalities are satisfied is the overlapping shaded region between the two lines.
I apologize for any confusion caused by my previous response. The correct representation of the solution region is the region bounded by both equations and can be expressed algebraically as:
-1/3x + 7 ≤ y ≤ -x + 3
So, you would shade the triangular region between the two lines, including its boundaries, in your graph.
in coordinates
To shade the solution region, we can determine the vertices of the triangle formed by the overlapping region of the two lines.
First, let's find the coordinates of the vertices:
1. Intersection point between the lines y = -1/3x + 7 and y = -x + 3:
To find this point, we can set the two equations equal to each other:
-1/3x + 7 = -x + 3
Multiply through by 3 to eliminate the fraction:
-x + 21 = -3x + 9
2x = 12
x = 6
Substitute x = 6 into either equation to find the y-coordinate:
y = -6 + 3
y = -3
The intersection point is (6, -3).
2. x-intercept of the line y = -1/3x + 7:
To find the x-intercept, we set y = 0 and solve for x:
0 = -1/3x + 7
1/3x = 7
x = 21
The x-intercept is (21, 0).
3. y-intercept of the line y = -x + 3:
To find the y-intercept, we set x = 0 and solve for y:
y = -0 + 3
y = 3
The y-intercept is (0, 3).
The shaded region is the triangle formed by connecting these three points: (6, -3), (21, 0), and (0, 3).