I'm having a hard time with graphing this problem. Could someone please help?
Graphing has never been one of my strongest points.
Graph the system of inequalities.
x+3y is less than or equal to 9, 3x+y is less than or equal to 9, x is greater than or equal to 0, y is greater than or equal to 0.
I thought that I had the graph but it turned out to be wrong.
Thanks.
first expression
below the line through (9,0) and (0,3)
below the line through (3,0) and (0,9)
Those happen to intersect at (4.5,4.5)
above the x axis and to the right of the y axis
Of course! I'm here to help you understand graphing inequalities. Let's start by graphing the individual inequalities one by one and then finding the overlapping region to get the solution to the system.
For the first inequality, x+3y ≤ 9, we need to find the line that represents this inequality. To do that, we'll set x+3y equal to 9 and draw a straight line. Since it's less than or equal to, we'll use a solid line to represent it.
To graph the line, we can start by finding its x and y-intercepts. Let's set x=0, then we have 3y ≤ 9, which implies y ≤ 3. So, we have a point at (0,3). Now, let's set y=0, then we have x ≤ 9. So, we have a point at (9,0). Plot these two points and draw a line through them.
Next, let's graph the second inequality, 3x+y ≤ 9. Following the same steps, we set 3x+y equal to 9 and draw a solid line. Let's find the intercept points: x=0 gives y ≤ 9, so we have a point at (0,9), and y=0 gives x ≤ 3, so we have a point at (3,0). Plot these points and draw a line through them.
Now we have two lines on the graph. The next steps involve shading to find the overlapping region. To identify which side to shade, we need to test a point in one of the inequalities not on the line. Let's take the point (0,0) and test it in the first inequality: 0+3(0) ≤ 9, which is true. So we shade the region below the first line.
Similarly, we can test the point (0,0) in the second inequality: 3(0)+0 ≤ 9, which is true. So we shade the region below the second line.
Finally, we need to consider the constraints x ≥ 0 and y ≥ 0. These represent the x and y axes. Since they are greater than or equal to, we draw solid lines on the x-axis and y-axis.
The region where all the shaded regions overlap is the solution to the system of inequalities. Shade this overlapping region.
I hope this explanation helps you to graph the system of inequalities correctly. If you have any further questions, please let me know!