Question 147092
Let x = the number of blueberry bushes
Let y = the number of peach trees


The amount of money she will spend on the bushes is 2.50x
The amount of money she will spend on the trees is 5.50y


The total amount she has to spend is 500, so the amount she spends on bushes plus the amount she spends on trees has to be less than or equal to 500.


{{{2.5x+5.5y<=500}}}


Graph {{{2.5x+5.5y=500}}}.  Since your original inequality includes 'or equal' make your graph a solid line.


Now, pick a point that is NOT on the line you just graphed. Since the line does not go through the origin, the point (0,0) is a good choice because it makes the arithmetic easy.


Substitute the coordinate values for the point you chose into the original inequality and do the arithmetic.  If the result is a true statement, shade in the half-plane containing the point you selected.  If the result is false, shade in the other side of the line.  In the case of your problem the result will be true, so the shaded area does, in fact, contain the origin (0,0).


For your problem, in practical terms, you also have other constraints, namely: {{{x>=0}}} and {{{y>=0}}}.  That's because she can't buy a negative number of either bushes or trees.  Therefore, you are only concerned with that part of your shaded area that is in the first quadrant of the coordinate system.


What this first quadrant shaded area represents is the area of feasibility.  In other words, any ordered pair (bushes,trees) with integer coordinates that lies within the area (including on the boundaries) is a possible combination that Linda could purchase from her supplier. Note that we had to put the 'integer coordinates' restriction in there because it is highly unlikely that a supplier will sell a fractional part of either a bush or a tree.


{{{drawing(600,600,-20,200,-20,200,

graph(600,600,-20,200,-20,200,(-2.5/5.5)x+(500/5.5)))}}}