How do we get to the vertex if this is the function: n(x)= |-1/3x + 1| =2 ??
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X = amount of the 25% copper alloy
y = amount of the 60% copper alloy
amounts are in kilograms
------------------------------------------------
The first equation is
x+y = 90
because the two alloy amounts (x and y) must combine to a total of 90 kg
We can solve for y to get
y = 90 - x
------------------------------------------------
Now onto the second equation.
x kg of the 25% alloy ---> 0.25*x kg of pure copper
y kg of the 60% alloy ---> 0.60*y kg of pure copper
In total, we have 0.25*x+0.60*y kilograms of pure copper
This is equal to 90*0.46 = 41.4 because the total amount of 90 kg alloy will have 46% copper
In short, the second equation is 0.25*x+0.60*y = 41.4
------------------------------------------------
Use the two equations (0.25*x+0.60*y = 41.4 and y = 90-x) to find x
I'm going to use substitution to plug in y = 90-x into the second equation
0.25*x+0.60*y = 41.4
0.25*x+0.60*( y ) = 41.4
0.25*x+0.60*( 90 - x ) = 41.4 <<<--- y has been replaced with 90-x
0.25*x+0.60*90 - 0.60*x = 41.4
0.25*x+54-0.60*x = 41.4
-0.35*x+54 = 41.4
-0.35*x = 41.4-54
-0.35*x = -12.6
x = -12.6/(-0.35)
x = 36
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If x = 36, then y is...
y = 90-x
y = 90-36
y = 54
------------------------------------------------
------------------------------------------------
In summary, we have
x = 36
y = 54
which means there are 36 kg of the 25% alloy
and 54 kg of the 60% alloy
(see attached for answer summaries)
y = amount of the 60% copper alloy
amounts are in kilograms
------------------------------------------------
The first equation is
x+y = 90
because the two alloy amounts (x and y) must combine to a total of 90 kg
We can solve for y to get
y = 90 - x
------------------------------------------------
Now onto the second equation.
x kg of the 25% alloy ---> 0.25*x kg of pure copper
y kg of the 60% alloy ---> 0.60*y kg of pure copper
In total, we have 0.25*x+0.60*y kilograms of pure copper
This is equal to 90*0.46 = 41.4 because the total amount of 90 kg alloy will have 46% copper
In short, the second equation is 0.25*x+0.60*y = 41.4
------------------------------------------------
Use the two equations (0.25*x+0.60*y = 41.4 and y = 90-x) to find x
I'm going to use substitution to plug in y = 90-x into the second equation
0.25*x+0.60*y = 41.4
0.25*x+0.60*( y ) = 41.4
0.25*x+0.60*( 90 - x ) = 41.4 <<<--- y has been replaced with 90-x
0.25*x+0.60*90 - 0.60*x = 41.4
0.25*x+54-0.60*x = 41.4
-0.35*x+54 = 41.4
-0.35*x = 41.4-54
-0.35*x = -12.6
x = -12.6/(-0.35)
x = 36
------------------------------------------------
If x = 36, then y is...
y = 90-x
y = 90-36
y = 54
------------------------------------------------
------------------------------------------------
In summary, we have
x = 36
y = 54
which means there are 36 kg of the 25% alloy
and 54 kg of the 60% alloy
(see attached for answer summaries)