Suppose that there are six universities and each will produce five mathematics Ph. D.s this year, and there are five colleges that will be hiring seven, seven, six, six, five math Ph. D.s, respectively. No college will hire more than one Ph. D. from any given university. Will all the Ph. D.s get a job? Explain.

No. If a collegue hire 7 math Ph.D.s, at least 2 will be from the same university.

The only way to have a job for all of them is that all colleges hire 6 math Ph.D.s, one from each university.

Step-by-step explanation:

If the condition is that no college will hire more than one Ph.D. from any given university, no college can hire more than 6 math Ph.D.s, as there are only 6 universities to choose from. If they hire seven, they have to repeat some university, violating the condition.

Then, the only way to have a job for all of them is that all colleges hire 6 math Ph.D.s, one from each university.

step-by-step explanation:

step 1: solve one of the equations for one of the variables. let's solve the first equation for y:

step 2: substitute that equation into the other equation, and solve for x.

step 3: substitute x = 4 x = 4 x=4 into one of the original equations, and solve for y.