Michael has $8 and wants to buy a combination of cupcakes and fudge to feed at least four siblings. Each cupcake costs $2, and each piece of fudge costs $1.
This system of inequalities models the scenario:
2x + y ≤ 8
x + y ≥ 4
Choose a different point in the solution set and interpret what it means in terms of the real-world context
One possible point in the solution set is (2, 2), which means Michael can buy 2 cupcakes and 2 pieces of fudge, spending a total of $6. This point satisfies both inequalities:
2x + y = 2(2) + 2(1) = 6 ≤ 8
x + y = 2 + 2 = 4 ≥ 4
Interpretation: If Michael buys 2 cupcakes and 2 pieces of fudge, he will spend $6 and have enough food to feed his four siblings. This is one possible combination of items that meets the requirements of the problem.
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Apologies, but I'm not able to continue the text for you.
Let's choose the point (x, y) = (2, 2) in the solution set of the given system of inequalities.
In this case, x = 2 represents the number of cupcakes Michael buys, and y = 2 represents the number of pieces of fudge he buys.
Interpreting this point in the real-world context, it means that Michael buys 2 cupcakes and 2 pieces of fudge. Since each cupcake costs $2 and each piece of fudge costs $1, Michael would spend 2 * $2 = $4 on cupcakes and 2 * $1 = $2 on fudge.
Therefore, he would spend a total of $4 + $2 = $6 to buy 2 cupcakes and 2 pieces of fudge, leaving him with $8 - $6 = $2.
To choose a different point in the solution set, we need to pick values for x and y that satisfy both inequalities given in the scenario.
Let's consider the point (x = 2, y = 6) as an example.
Plugging in these values into the inequalities:
2x + y ≤ 8
2(2) + 6 ≤ 8
4 + 6 ≤ 8
10 ≤ 8 (This is not true)
x + y ≥ 4
2 + 6 ≥ 4
8 ≥ 4 (This is true)
Since the first inequality is not true, the point (x = 2, y = 6) is not a solution to the system of inequalities.
Therefore, let's choose another point that does satisfy both inequalities.
Let's consider the point (x = 1, y = 3).
Plugging in these values into the inequalities:
2x + y ≤ 8
2(1) + 3 ≤ 8
2 + 3 ≤ 8
5 ≤ 8 (This is true)
x + y ≥ 4
1 + 3 ≥ 4
4 ≥ 4 (This is true)
Since both inequalities are true, the point (x = 1, y = 3) is a solution to the system of inequalities.
Interpretation in terms of the real-world context:
The point (x = 1, y = 3) means that Michael can buy 1 cupcake and 3 pieces of fudge, spending a total of $7 (2*1 + 1*3 = $5 + $3 = $7). This combination would satisfy the given conditions for feeding at least four siblings (he would have 1 cupcake and 3 pieces of fudge to distribute among them).