+43 Hi guys! I am not sure how to solve this type of question:
\(\text{Given that }-3 \le 7x + 2y \le 3\text{ and }-4 \le y - x \le 4,\text{ what is the maximum possible value of }x + y?\)
+43 Worked out how to do it myself - sorry for any inconvenience.
To solve this problem, we need to get x+y expressed in an inequality so we can work out the maximum possible value. In order to do that, we need \(x\&y\) to have an equal coefficient.
Let's multiply the first inequality by 2: \((-3\le7x+2y\le3)\times2=-6\le14x+4y\le3.\)
Now we need to multiply the second inequality by 5: \((-24\le y-x\le4)\times5=-20\le5y-5x\le20.\)
Now we can add both inequalities. Note that we can only do this if the signs on the inequalities are the same, but, since they are, we are all good.
\((-20\le5y-5x\le20)+(-6\le14x+4y\le3)=-26\le9x+9y\le26.\)
Now we can isolate x+y by dividing the expression by 9: \((-26\le9x+9y\le26)\times9=-\dfrac{26}{9}\le x+y\le\dfrac{26}{9}.\)
Since the highest value of the inequality is \(\dfrac{26}{9}\), the maximum possible value of x+y is \(\dfrac{26}{9}.\)
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