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: (-3,-7), (-7,-3), (3,7), and (7,3) Maximum: 43 Minimum: -43
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.
y≤ x+4 & (I) y≥ x-4 & (II) y≤- x+10 & (III) y≥- x-10 & (IV)
Let's consider given inequalities in pairs.
Inequality I & II
We will start by changing the inequality symbols in the first two inequalities to an equals sign. Then we can highlight slopes m and y-intercepts b in boundary line equations to draw the lines.
Information
Inequality (I)
Inequality (II)
Given Inequality
y ≤ x+4
y ≥ x-4
Boundary Line Equation
y = x+4
y = x-4
y=mx+b
y=1x+4
y=1x+(-4)
Now that we know the slopes and y-intercepts, let's use these to draw the boundary lines. Notice that the inequalities are non-strict, which means that the lines will be solid.
To complete the graph, we will test a point that does not lie on either boundary line and decide which region we should shade. Let's choose ( 0, 0). We will substitute this point in the given inequalities and simplify. If the substitution creates a true statement, we shade the same region as the test point. Otherwise, we shade the opposite region.
Information
Inequality (I)
Inequality (II)
Given Inequality
y≤ x+4
y≥ x-4
Substitute (0,0)
0? ≤ 0+4
0? ≥ 0-4
Simplify
0≤ 4 ✓
0≥ -4 ✓
Since for both inequalities we got a true statement, we will shade the regions containing our test point.
Inequality III & IV
Again we will start by changing the inequality symbols in the last two inequalities to an equals sign. Then we will highlight slopes m and y-intercepts b in boundary line equations to draw the lines.
Information
Inequality (III)
Inequality (IV)
Given Inequality
y ≤ - x+10
y ≥ - x-10
Boundary Line Equation
y = - x+10
y = - x-10
y=mx+b
y=-1x+10
y=-1x+(-10)
Now that we know the slopes and y-intercepts, let's use these to draw the boundary lines. Notice that the inequalities are non-strict, which means that the lines will be solid.
To complete the graph, we will test a point that does not lie on either boundary line and decide which region we should shade. Let's choose ( 0, 0). We will substitute this point in the given inequalities and simplify. If the substitution creates a true statement, we shade the same region as the test point. Otherwise, we shade the opposite region.
Information
Inequality (III)
Inequality (IV)
Given Inequality
y≤ - x+10
y≥ - x-10
Substitute (0,0)
0? ≤- 0+10
0? ≥- 0-10
Simplify
0≤ 10 ✓
0≥ -10 ✓
Since for both inequalities we got a true statement, we will shade the regions containing our test 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.
( -3, -7), ( -7, -3), ( 3, 7), and ( 7, 3)
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+9y, we will determine its maximum and minimum values. Let's start with the vertex ( -3, -7).
For the vertex ( -3, -7), the value of the function is -33. We can determine the values of the function for the other vertices in the same way.
Vertex
f(x,y)=-10x+9y
Value
( -3, -7)
f( -3, -7)=-10( -3)+9( -7)
f(-3,-7)=-33
( -7, -3)
f( -7, -3)=-10( -7)+9( -3)
f(-7,-3)=43
( 3, 7)
f( 3, 7)=-10( 3)+9( 7)
f(3,7)=33
( 7, 3)
f( 7, 3)=-10( 7)+9( 3)
f(7,3)=-43
Looking at the table, we can see that the maximum value of given function is 43 and it is reached at (-7,-3). The minimum value is -43 and it occurs at (7,3).
