Graph the given system and determine the vertices of the overlapping region.
Garph:
Vertices: (- 3,1.5), (5,7.5), (0,- 6)
Practice makes perfect
Our first step in finding the vertices is to graph the system and determine the overlapping region. Graphing a single inequality involves two main steps.
Plotting the boundary line.
Shading half of the plane to show the solution set.
For this exercise, we need to do this process for each of the inequalities in the system.
- 3x+4y≤ 15 & (I) 2y+5x>- 12 & (II) 10y+60≥ 27x & (III)
Let's begin!
Boundary Lines
We can tell a lot of information about the boundary lines from the inequalities given in the system.
Exchanging the inequality symbols for equals signs gives us the boundary line equations.
Observing the inequality symbols tells us whether the inequalities are strict.
Writing the equation in slope-intercept form will help us highlight the slopes m and y-intercepts b of the boundary lines.
Let's find each of these key pieces of information for the inequalities in the system.
Information
Inequality (I)
Inequality (II)
Inequality (III)
Given Inequality
- 3x+4y≤15
2y+5x>- 12
10y+60≥27x
Boundary Line Equation
- 3x+4y=15
2y+5x=- 12
10y+60=27x
Solid or Dashed?
≤ ⇒ Solid
> ⇒ Dashed
≥ ⇒ Solid
y= mx+ b
y= 3/4x+ 15/4
y= -5/2x+( - 6)
y= 2.7x+( - 6)
Great! With all of this information, we can plot the boundary lines. Let's do it one at a time.
Inequality I
To draw the first boundary line we will plot the y-intercept and then use the slope to find another point.
To complete the graph, we will test a point that is not on the line and decide which region we should shade. Let's test the point ( 0, 0). If the point satisfies the inequality, we will shade the region that contains the point. Otherwise, we will shade the other region.
Thus, we have to shade the region containing the point.
Combining the Inequality Graphs
Finally, we can draw the graphs of the inequalities on the same coordinate plane.
Now that we can see the overlapping region, let's highlight the vertices.
Looking at the graph, we can determine the first vertex.
(0,- 6)
To find the second vertex, we will have to determine the point of intersection of the lines - 3x+4y=15 and 10y+60=27x. We can do it by solving the system of equations related to these lines.
- 3x+4y=15 & (I) 10y+60=27x & (II)
Since there is no isolated variable, we will use the Elimination Method.
The second vertex is (5,7.5). Now, we will find the point of intersection of the lines - 3x+4y=15 and 2y+5x=- 12, by solving the following system.
- 3x+4y=15 & (I) 2y+5x=- 12 & (II)
Again, let's use the Elimination Method.
