Liam is setting up folding chairs for a meeting. If he arranges the chairs in 8 rows of the same length, he has 9 chairs left over. If he arranges the chairs in 6 rows of that same length, he has 19 left over. How many chairs does Liam have? please answer quickly
8x+ 9 = 6x + 19
2x = 10
x = the number of chairs per row. or 5
To get total number of chairs; 8(5) + 9 should get you the same answer as 6(5) +19
let the number of longer rows be x
total = 8x + 9
let the number of shorter rows be y
total = 6y + 19
8x + 9 = 6y + 19
8x = 6y + 10
4x = 3y + 5
x = (3y+5)/4
so any ordered pair of whole numbers satisfying this would work
e.g. (2,1) would work, of course the total of chairs would change
check: if x = 2, y = 1
total of chairs would be 8x + 9 or 25
(notice I get the same total from 6y + 19 )
for longer rows: 2 x 8 = 16 , leaving 9
for shorter rows: 1 x 6 = 6, leaving 19
other examples (5 , 5), (8, 9), .....
(since the slope of 4x = 3y + 5 is 4/3, we can just add 4 to y and 3 to x )
To find the number of chairs Liam has, we can set up a system of equations based on the information provided.
Let's assume that each row has x chairs.
According to the first statement, if Liam arranges the chairs in 8 rows of the same length, he has 9 chairs left over. This can be represented as:
8x + 9 = Total number of chairs -- (Equation 1)
Similarly, according to the second statement, if Liam arranges the chairs in 6 rows of the same length, he has 19 chairs left over:
6x + 19 = Total number of chairs -- (Equation 2)
We now have a system of equations:
8x + 9 = Total number of chairs
6x + 19 = Total number of chairs
To solve this system of equations, we can equate the two equations:
8x + 9 = 6x + 19
Simplifying the equation, we get:
2x = 10
Dividing both sides by 2, we find:
x = 5
Now that we know the value of x, we can substitute it back into either equation to find the total number of chairs. Let's use Equation 1:
8x + 9 = Total number of chairs
8(5) + 9 = Total number of chairs
40 + 9 = Total number of chairs
49 = Total number of chairs
Therefore, Liam has a total of 49 chairs.
Please note that it's important to solve step-by-step and double-check to ensure the accuracy of the answer.