teaching me to solve ---12. Consider the following simultaneous equations: xy² + xyz = 91, xyz - y²z = 72, where x, y, and z are positive integers. Find the maximum value of xz.

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We have a system of two simultaneous equations with three positive integer variables x, y, and z. The first equation is xy squared plus xyz equals 91. The second equation is xyz minus y squared z equals 72. Our goal is to find the maximum value of xz. The first step is to factor out common terms from both equations. In the first equation, we can factor out xy from both terms, giving us xy times the quantity y plus z equals 91. In the second equation, we can factor out yz from both terms, giving us yz times the quantity x minus y equals 72. This factoring will help us identify the constraints on our variables. Now we need to find the common factors of 91 and 72 to determine the possible values of y. The factors of 91 are 1, 7, 13, and 91. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Looking at both lists, the only common factor is 1. Therefore, y must equal 1. Now we substitute y equals 1 into our factored equations. The first equation becomes x times 1 times the quantity 1 plus z equals 91, which simplifies to x times the quantity 1 plus z equals 91. The second equation becomes 1 times z times the quantity x minus 1 equals 72, which simplifies to z times the quantity x minus 1 equals 72. We now have a system of two equations with two unknowns, x and z. Now we solve for x and z by testing the factors of 91. Since x must be greater than 1, we test x equals 7, 13, and 91. For x equals 7: substituting gives us z equals 12, and checking the second equation confirms this works. So xz equals 84. For x equals 13: substituting gives us z equals 6, which also satisfies both equations. So xz equals 78. For x equals 91: this gives z equals 0, which is invalid since z must be positive. Therefore, the maximum value of xz is 84.