Question 1167407
here's what i get and what i think is wrong with this problem.


let y equal the number of chairs needed.


let x equal the number of rows.


when you have 8 chairs in a row, then:


8x + 7 = y


this means you multiply 8 * x and get 7 less chairs than you need, so you have to add 8 to 8x.


when you have 7 chairs in a row, then:


7x - 2 = y


this means you multiply 7 * x and get 2 more chairs than you need, so you have to subtract 2 from 7x.


with 8 chairs in a row, multiplied by x, you need 7 more chairs to equal y.
with 7 chairs in a row, multiplied by x, you need 2 less chairs to equal y.


your two equations that you need to solve simultaneously are:


8x + 7 = y
7x - 2 = y


subtract the second equation from the first to get:


x + 9 = 0


solve for x to get:


x = -9


that's not good because x has to be greater than zero.


but, it does give you a solution, even though the solution is the wrong sign.


so, i said:


what makes the solution positive?


the answer was to assume the reverse.


8x + 7 = y becomes 8x - 7 = y
8x - 2 = y becomes 8x + 2 = y


solve these two equations simultaneously and i get:


8x - 7 = y
7x + 2 = y


subtract the second equation form the first and i get:


x - 9 = 0


solve for x to get x = 9.


the answer appears to be valid, but the conditions are reversed.


8 * 9 - 7 = 72 - 7 = 65
7 * 9 + 2 = 63 + 2 = 65


this says the number of chairs i needed was 65.


with 8 chairs in a row, i got 72 chairs which was 7 more than i needed.
with 7 chairs in a row, i got 63 chairs which was 2 less than i needed.


the number of chairs needed was between 30 and 70.
65 is between 30 and 70, so that requirement checked out ok.


with my original assumption, based on what i thought the problem was stating, i did not get a solution.


the following graphs show what i mean.


<img src = "http://theo.x10hosting.com/2020/101601.jpg" >


<img src = "http://theo.x10hosting.com/2020/101602.jpg" >


in the first graph, the equation graphed are y = 8x + 7 and y = 7x - 2.


as you can see there is no solution.


in the second graph, the equation graphed are y = 8x - 7 and y = 7x + 2.


as you can see there is a solution at x = 9.


the solution, in both cases, had to be between y = 30 and y = 70, as stated in the problem statement.


the only time i got a solution was when y = 8x - 7 and y = 7x = 2.


what did these equations mean?


since y = the number of chairs needed and x = the number of rows, then:


y = 8x - 7 means that the number of chairs needed is equal to 8 * the number of rows - 7 which means that you get 7 more chairs than you need.
if you add 7 to both sides of this equation, you get y + 7 = 8x.
this says that 8 chairs in a row times the number of rows gets you 7 more chairs than you need.


y = 7x + 2 means that the number of chairs needs is equal to 7 * he number of rows + 2 which means that you get 2 less chairs than you need.
if you subtract 2 from both sides of this equation, you get y - 2 = 8x.
this says that 7 chairs in a row time the number of rows gets you 2 less chairs than you need.


this is opposite what the problem statement says, which is: "If he puts 8 chairs in one row there would be 7 chairs short . If he put 7 in a row there would be 2 left."


what you actually get is the opposite, which is: "If he puts 8 chairs in one row there would be 7 chairs left. If he put 7 in a row there would be 2 short."


7 chairs left means he gets 7 more chairs than he needed.
2 chairs short means he gets 2 less chairs than he needed.


anyway, this is what i think.


i believe the solution to your problem is that the number of rows is 9.


with 9 rows:


9 * 8 = 72 and 9 * 7 = 63
72 is 7 rows more than you need.
63 is 2 rows less than you need.