Let's begin by defining the variables for the situation.

x : Number of packages of paint brushes y : Number of boxes of colored pencils

We know that Mr. Williams wants to but at least

2

packages of each supply. Therefore, first two constraints of the system can be written as follows.

{x \geq 2 y \geq 2 (I) (II)

We have more information to write another constraint for the cost of the supplies. We can write the third constraint by organizing the given information on a table.

Verbal Expression	Algebraic Expression
Cost of $x$ packages of paint brushes ( $$$ )	$4.75 x$
Cost of $y$ boxes of colored pencils ( $$$ )	$6.50 y$
Total cost of the supplies must be less than or equal to $$ 50 .$	$4.75 x + 6.50 y \leq 50$

Now we have three inequalities to write a system.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ x \geq 2 y \geq 2 4.75 x + 6.5 y \leq 50 (I) (II) (III)

To graph the system, we will begin by determining the boundary lines of Inequality (I) and (II). Boundary line can be determined by replacing the inequality symbol with the equals sign.

I : II : \underline{Inequality} x \geq 2 y \geq 2 \underline{Boundary Line} x = 2 y = 2

Boundary Line (I) is a vertical line that passes through the point

(2, 0)

and Boundary Line (II) is a horizontal line that passes through the point

(0, 2) .

Because of the non-strict inequalities, both of them will be solid. They will also be bounda by the axes because the number of supplies cannot be negative. Let's graph them!

Inequality (I) states that the points with $x -$ coordinates greater than or equal to $2$ are included in the solution set. Therefore, we will shade the region to the right of Boundary Line (I). Thinking in the same way, we can shade above Boundary Line (II). Let's do it!

Next, we will graph Inequality (III).

\underline{Inequality III} 4.75 x + 6.5 y \leq 50 \underline{Boundary Line III} 4.75 x + 6.5 y = 50

As we can see, the boundary line is in standard form. Therefore, it would be a better option to find its intercepts to graph it. We will substitute

y = 0

for the

x -

intercept and

x = 0

for the

y -

intercept.

	$4.75 x + 6.5 y = 50$
Operation	$x -$ intercept	$y -$ intercept
Substitution	$4.75 x + 6.50 = 50$	$4.750 + 6.5 y = 50$
Calculation	$x = 10.53$	$y = 7.69$
Point	$(10.53, 0)$	$(0, 7.69)$

Now we can plot the intercepts and connect them with a line segment. The boundary line will be solid because of the non-strict inequality.

Next, we will decide which region we should shade by testing the point

(0, 0) .

4.75 x + 6.5 y \leq 50

SubstituteII

$x = 0$ , $y = 0$

4.75 (0) + 6.5 (0) \leq ? 50

ZeroPropMult

Zero Property of Multiplication

0 \leq 50

Since the point satisfies the inequality, region that contains the point will be shaded.

The overlapping section of the graph above represents the feasible region.

The number of supplies is whole numbers. Therefore, the points with whole number coordinates in the overlapping sections are the solutions to the system. Let's choose three of them!

Number of Packages of Paint Brushes	Number of Boxes of Colored Pencils
$3$	$3$
$4$	$3$
$4$	$4$

Exercise