Graph the given system and determine the vertices of the overlapping region.
Graph:
Vertices: (2,- 1), (5,8), (- 7,8)
Practice makes perfect
Our first step in finding the vertices is to graph the system and determine the overlapping region.
y≥ 3x-7 & (I) y≤ 8 & (II) x+y>1 & (III)
Inequaliy I
To graph the inequality, we will have to determine the boundary line first. We can do it by changing the inequality symbol to an equals sign.
Inequality:& y≥ 3x-7
Boundary Line:& y=3x-7
Since this equation is in slope-intercept form, we can determine its slope m and y-intercept b to draw the line.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= 3x+(- 7)
Now that we know the slope and y-intercept, let's use these to draw the boundary line. Notice that the inequality is not strict, which means that the line will be solid.
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.
Since the point satisfies the inequality, we will shade the region that contains the point.
Inequality II
The inequality y≤ 8 tells us that all coordinate pairs with a y-coordinate that is less than or equal to 8 will be in the solution set of the inequality. In this case the boundary line will be horizonatl. It will also be solid, since the inequality is non-strict.
Inequality III
Let's graph the third inequality. We will begin by identifying the boundary line.
Inequality:& x+y>1
Boundary Line:& x+y=1
Let's rewrite this equation in slope-intercept form.
Below we highlight the slope m and the y-intercept b.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= - 1x+1
Let's graph the boundary line. Notice that the inequality is strict, which means that the line will be dashed.
Since the point does not satisfy the inequality, we will shade the region opposite the point.
Combining the Inequality Graphs
Let's 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 identify the vertices.
(2,- 1), (5,8), (- 7,8)
Extra
Extra
If you cannot determine a vertex by looking at the graph, you have to solve a system of equations. Each vertex is also a point of intersection of two boundary lines. Thus, to find it, we have to solve the system of equations related to these lines. Let's find the vertex created by the lines x+y=1 and y=3x-7.
x+y=1 & (I) y=3x-7 & (II)
Since the y-variable is isolated in the second equation, we willl use the Substitution Method.
