Graph the given system and determine the vertices of the overlapping region.
Graph:
Vertices: (3.5,8), (- 4,8), (0.5,2)
Practice makes perfect
Our first step in finding the vertices is to graph the system and determine the overlapping region.
y≥ 2x+1 & (I) y≤ 8 & (II) 4x+3y≥ 8 & (III)
Inequality 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≥ 2x+1
Boundary Line:& y=2x+1
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= 2x+1
Now that we know the slope and y-intercept, let's use these to draw the boundary line. Notice that the inequality is non-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 does not satisfy the inequality, we will shade the region that does not contain 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. Because the inequality is non-strict, the boundary line will be solid.
Inequality III
To write the equation of the boundary line for the third inequality, we will first isolate the y-variable.
Next, we will write the equation of the boundary line.
Inequality:& y≥-4/3x+8/3
Boundary Line:& y=-4/3x+8/3
Below we highlight the slope m and the y-intercept b.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= -4/3x+8/3
Let's draw the boundary line. Notice that the inequality is non-strict, which means that the line will be solid.
Since the point does not satisfy the inequality, we will shade the region that does not contain 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 see that the line y=8 intersects the lines y=2x+1 and y=- 43x+ 83 at the points (3.5,8) and (- 4,8), respectively. These are two of the vertices.
(3.5,8) and (- 4,8)
To find the third vertex, we will have to determine the point of intersection of the lines y=2x+1 and y=- 43x+ 83. We can do it by solving the system of equations related to these lines.
y=2x+1 & (I) y=- 43x+ 83 & (II)
Since the y-variable is already isolated, we will use the Substitution Method.
