Jimmy is going to the store to buy candles. Small candles cost $3.50 and large candles cost $5.00. He needs to buy at least 20 candles, and he cannot spend more than $80.

Question

Let x represent the number of small candles and y represent number of large candles. (a) Write a system of linear inequalities to represent the situation. (b) Identify 2 solutions to the system of linear inequalities. (c) Verify both solutions are valid by showing work to verify they satisfy both linear inequalities.

DO I GRAPH THIS?

Answer 1

No, you do not need to graph this. Instead, do the following:

(a) The number of small candles + the number of large candles = x + y >= 20 (1), and the cost of small candles * the number of small candles + the cost of large candles * the number of large candles = 3.5*x + 5*y <= 80 (2).

(b) The 2 solutions can be obtained by solving this linear system of equations, instead of inequalities, with A being <= 80 such that both x

and y are integers, since you can't buy a fraction of a candle:

x + y = 20 --> multiply both sides by +5 --> 5*x + 5*y = 100

3.5*x + 5*y = A --> multiply both sides by - 1 --> -3.5*x - 5*y = -A

Add equations (2) and (3) to eliminate y & solve for x --> 1.5*x = 3*x/2 = 100 - A.

So, 3*x = 200 - 2*A, but for x to be an integer, this has to be divisible by 3, so

instead of solving for x, pick the following 2 solutions:

(x = 20, y = 0) and (x = 20, y = 2)

(c) Check 2 solutions by plugging x and y into inequalities (1) & (2) from part (a):

For (x = 20, y = 0), x + y = 20 + 0 = 20 which is >= 20 and 3.5*x + 5*y =

3.5*(20) + 5*(0) = 7*20/2 = 7*10 = 70 which is <= 80, so this is a valid solution.

For (x = 10, y = 10), x + y = 20 + 2 = 22 which is >= 20 and 3.5*x + 5*y =

3.5*(20) + 5*(2) = 7*20/2 + 10 = 7*10 + 10 = 70 + 10 = 80 which is <= 80, so this is also a valid solution.