b = price of one belt
h = price of one hat
prices are in dollars
1 belt costs b dollars where b is simply a placeholder for a number. So this means 6 belts cost 6*b or 6b dollars. Eight hats cost 8*h or 8h dollars. In total, the whole lot is 6b+8h dollars. This batch costs $140, so 6b+8h = 140 is the first equation. The second equation is 9b+6h = 132 for similar reasons as the first equation.
So we have this system of equations
6b+8h = 140
9b+6h = 132
Let's use elimination to solve this system. If we wanted to eliminate h, then adding 8h+6h gets us 14h when we want 0h instead. Recall that 0 added to anything does not change the number. So if we can get a sum of 0h then the h terms go away.
What we can do is two things:
* Multiply both sides of equation (1) by -3
* Multiply both sides of equation (2) by 4
This will turn the original system
6b+8h = 140
9b+6h = 132
Into this system
-18b-24h = -420
36b+24h = 528
Add up the equations
-18b-24h = -420
36b+24h = 528
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18b+0h = 108
We now have 18b = 108 which solves to b = 6 after dividing both sides by 18
After figuring out the value of b, we use it to find the value of h
6b+8h = 140
6*6+8h = 140 ... plug in b = 6
36+8h = 140
36+8h-36 = 140-36
8h = 104
8h/8 = 104/8
h = 13
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Answers:
The cost of one belt is $6
The cost of one hat is $13