Graph the given system and determine the vertices of the overlapping region. Substituting these vertices into the function will help you find the maximum and minimum values.
Graph:
Vertices: (6,3), (-8,10), and (-8,-18) Maximum: 42 Minimum: -140
Practice makes perfect
Our first step in finding the maximum and minimum values of the given function is to graph the system and determine the vertices of the overlapping region. Substituting these vertices into the equation, we will find these values.
x≥ -8 & (I) 3x+6y≤36 & (II) 2y+12≥3x & (III)
Inequality I
The inequality x ≥ -8 describes all values of x that are greater than or equal to -8. This means that every coordinate pair with an x-value that is greater than or equal to -8 will be included in the shaded region. Notice that the inequality is not strict, so the boundary line will be solid.
Inequality II
Let's start by changing the inequality symbol in the second inequality to an equals sign.
Inequality:& 3x+6y≤36
Boundary Line:& 3x+6y=36
The second step will be to rewrite the equation in the slope-intercept form.
Now, we can determine its slope m and y-intercept b to draw the line.
Slope-Intercept Form:& y= mx+b
Boundary Line:& y= -1/2x+(-6)
Let's use the slope and y-intercept 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 satisfies the inequality, we will shade the region that contains the point.
Inequality III
Let's again start by changing the inequality symbol in the last inequality to an equals sign.
Inequality:& 2y+12≥3x
Boundary Line:& 2y+12=3x
The second step will be to rewrite the equation in the slope-intercept form.
Since the 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= 3/2x+(-6)
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 satisfies the inequality, we will shade the region that contains 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 write down all four vertices.
( 6, 3), ( -8, 10), and ( -8, -18)
Having all the vertices we can find the maximum and minimum of the given function.
Finding the Maximum and Minimum Values
Substituting the vertices into the given function, f(x,y)=10x-6y, we will determine its maximum and minimum values. Let's start with the vertex ( 6, 3).
For the vertex ( 6, 3), the value of the function is 42. We can determine the values of the function for the other vertices in the same way.
Vertex
f(x,y)=10x-6y
Value
( 6, 3)
f( 6, 3)=10( 6)-6( 3)
f(6,3)=42
( -8, 10)
f( -8, 10)=10( -8)-6( 10)
f(-8,10)=-140
( -8, -18)
f( -8, -18)=10( -8)-6( -18)
f(2,9)=28
Looking at the table, we can see that the maximum value of given function is 42 and it is reached at (6,3). The minimum value is -140 and it occurs at (-8,10).
