We need to graph the following system of inequalities:
5 x + 7 y >= 7
- 7 x + 4 y > -12
I see a letter "y"in the first inequality, please provide the image. Got it!
Then, we start by plotting the associated "boundary lines"for each inequality:
We study first 5 x + 7 y = 7 (notice we are using just an equal sign instead of the inequality symbol because we are looking for the LINE that divides the plane)
Then we solve for y in the equation to guide us:
subtract 5x from both sides:
7 y = - 5 x + 7
divide by 7 to isolate the y:
y = - 5/7 x + 7/7
y = -5/7 x + 1
Then we look for two points to be able to draw this division line:
when x is 0 the equation becomes:
y = -5/7 (0) + 1 = 1
Then (0, 1) is a point of the line.
and we find another point by using for example x = 7
y = - 5/7 (7) + 1 = -5 + 1 = -4
therefore the two points we use to draw this division line are: (0, 1) and (7, -4)
We then have to decide if we draw a SOLID or a DOTTED line for this boundary line. Since our inequality contains the equal sign as well, we opt for a SOLID line to represent the boundary line.
Now, we go back to the original inequality to decide which region of the plane we want to highlight as solution of the first inequality. We can use the point (0, 0) which is outside this line and see if it works for the inequality:
5 (0) + 7 (0) = 0 Is this larger than or equal to 7 as the original inequality requested? The answer is no, so we know that we need to select the hal plane that DOESNOT contain the point (0, 0).
Allow me a few minutes to darw this first inequality on the plane and upload the image here:
Now we work on the same process to draw the boundary line associated with the second inequality and then deciding which half plane to highlight:
The second inequality is: - 7 x + 4 y > -12
then, the associated equation will be:
- 7 x + 4 y = - 12
once again we solve for y:
4 y = 7 x - 12
divide by 4 both sides:
y = 7/4 x - 3
This is the boundary line for the second inequality.
Now we ask ourselves if we need to draw a SOLID line or a DOTTED line to represent this boundary line. The answer is: We draw A DOTTED line, because this inequality symbol does not contain an equal sign, its is just the symbol >
We find two points on the plane that allow us to draw the boundary line. We select to use easy values for x, like x = 0 that gives:
y = 7/4 (0) - 3 = -3
Then the point (0, -3) is one point we are going to use to draw the boundary line.
For the other point, we use for example x = 4
y = 7/4 (4) - 3
y = 7 - 3 = 4
Then the point (4, 4) is on our boundary line.
We then iuse the point (0, 0) to decide which half plane we need to higlight. We use it in the original inequality to see if it holds true:
- 7 (0) + 4 (0) = 0 Is this larger than -12 as the inequality requires? The answer is YES, so we highlight that section of the plane.
Let me add now the plot of both inequalities overlapped:
You can see the new "boundary line" for the second inequality plotted in green color and with dotted line, and the half plane that is the selected one highlighted in green.
The OVERLAPPING region (color brownish) is the actual solution to the system of inequalities.
If they ask you for the coordinates of ONE point that is solution of the system of inequalities, we can pick any point in the brown region. For exaple the point (2, 2) which is cklearly there.
We can even check that such point is solution of the system by using it in each inequality and observing that it gives true answers to both.