Begin by writing a system of inequalities that represents the situation. Then, graph the system. You should remember that at least means greater than or equal to and no more than means less than or equal to.
Possible Combination: 15 batches of cookies and 6 batches of cupcakes
Practice makes perfect
Let x be the number of cookie batches and y be the number of cupcake batches. We will first write a system of inequalities that represents the situation. We should remember that at least means greater than or equal to and no more than means less than or equal to.
Verbal Expression
Algebraic Expression
Number of cookies after x batches
15 x
Number of cupcakes after y batches
12 y
Total number of goods is greater than or equal to 120
15 x+ 12 y ≥ 120
Total number of goods is less than or equal to 360
15 x+ 12 y ≤ 360
Thus, we have the inequalities to write the system
15x+12y≥ 120 & (I) 15x+12y≤ 360 & (II)
To graph the system, we will first determine the boundary lines of the inequalities that form it. The boundary line can be determined by replacing the inequality symbol with the equals sign.
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&Inequality & Boundary Line
I: &15x+12y ≥ 120 &15x+12y = 120
II: &15x+12y ≤ 360 &15x+12y = 360
Next, we will determine the intercepts of both boundary lines to graph them. Let's begin by finding the x-intercept of the first boundary line. To do that we will substitute 0 for y into the equation and solve it for x.
The x-intercept of the first boundary line is the point (8,0). The x- and y-intercepts of the lines can be found proceeding in the same way.
15x+12y=120
15x+12y=360
Operation
x-intercept
y-intercept
x-intercept
y-intercept
Substitution
15x+12( 0)=120
15( 0+12y=120
15x+12( 0)=360
15( 0+12y=360
Calculation
x=8
y=10
x=24
y=30
Point
(8,0)
(0,10)
(24,0)
(0,30)
Now that we know the intercepts, we can plot them and draw the lines that passes through them. Notice that the number of batches cannot be negative, so they will be bound by the axes. They will also be solid because both inequalities are non-strict.
Next, we will test an arbitrary point to decide which regions we should shade. Let's first test the point (0,0) for the first inequality.
This time, the point satisfied the inequality. Therefore, we will shade the region that contains the point.
The overlapping section of the graph above is the solution set of the system.
Looking at the graph, we can say that Rebecca could make 15 batches of cookies and 6 batches of cupcakes. Notice that there are more than one combination.
