Exercise 17 Page 161

Our first step in finding the maximum value of the given equation is to graph the given system and determine the vertices of the overlapping region. Substituting these vertices into the equation, we will find the maximum value.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 3 x + y \leq 7 x + 2 y \leq 9 x \geq 0, y \geq 0 (I) (II) (III)

Inequality I

To write the equation of the boundary line for the first inequality, we will first isolate the

y -

variable.

3 x + y \leq 7

SubIneq

$LHS - 3 x \leq RHS - 3 x$

y \leq - 3 x + 7

We can write the equation of the boundary line by changing the inequality symbol to an equals sign.

Inequality : Boundary Line : y \leq - 3 x + 7 y = - 3 x + 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 : Boundary Line : y = m x + b y = - 3 x + 7

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.