Let p,n,m be numbers such that pn=7 and 7p−7n=126 and m=p+n. Find the value of m.

Step 1: We have two equations, pn = 7 and 7p - 7n = 126.

Step 2: We can simplify the second equation by dividing every term by 7, which gives us p - n = 18.

Step 3: Now we have two new equations, pn = 7 and p - n = 18.

Step 4: We can solve these two equations simultaneously. Let's express p from the first equation: p = 7/n.

Step 5: Substitute p = 7/n into the second equation: 7/n - n = 18.

Step 6: Multiply every term by n to get rid of the denominator: 7 - n^2 = 18n.

Step 7: Rearrange the equation to form a quadratic equation: n^2 + 18n - 7 = 0.

Step 8: Solve the quadratic equation for n. The solutions are n = -18.07 and n = 0.39.

Step 9: Substitute n = -18.07 into the first equation pn = 7 to find the corresponding p. We get p = -0.39.

Step 10: Substitute n = 0.39 into the first equation pn = 7 to find the corresponding p. We get p = 18.07.

Step 11: We are asked to find m = p + n. So, m = -0.39 - 18.07 = -18.46 and m = 18.07 + 0.39 = 18.46.

Step 12: Since m must be a positive number, we discard the negative solution. So, m = 18.46.