Question 1097569
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"A chemical manufacturer wishes to fill an order for 700 gallons of a 24% acid solution" means they want 700*0.24 = 168 gallons of pure acid


Let 
x = amount of the 20% acid solution
y = amount of the 30% acid solution
both amounts are in gallons


Since we can only use the 20% or the 30% solution, this means that x and y must add to the total of 700

So we have the equation
x+y = 700


Now solve for y. Subtract x from both sides
x+y = 700
x+y-x = 700-x
y = 700-x
Keep this equation in mind. We'll use this equation later


If we use x gallons of the 20% solution, then we have 0.20*x gallons of pure acid. If we use y gallons of the 30% solution, then we have 0.30*y gallons of pure acid. Combined we have 0.20*x+0.30*y gallons of pure acid. Recall that above we want the target amouunt of total pure acid to be 168 gallons. That is how we're able to get the equation below


0.20*x + 0.30*y = 168


Now use the equation y = 700-x to perform substitution
0.20*x + 0.30*y = 168
0.20*x + 0.30*( y ) = 168
0.20*x + 0.30*( 700 - x ) = 168 ... y has been replaced with 700-x


With y gone from the equation, we can now solve for x
0.20*x + 0.30*( 700 - x ) = 168
0.20*x + 0.30*( 700 ) + 0.30*( - x ) = 168
0.20*x + 210 - 0.30x = 168
0.20x - 0.30x + 210 = 168
-0.10x + 210 = 168
-0.10x + 210-210 = 168-210
-0.10x = -42
-0.10x/(-0.10) = -42/(-0.10)
x = 420


Use this x value to find y (again use y = 700-x)
y = 700-x
y = 700-420 ... replace x with 420
y = 280


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In summary we found that 
<font color=red>x = 420</font>
<font color=blue>y = 280</font>


So we should use <font color=red>420 gallons</font> of the 20% solution, and <font color=blue>280 gallons</font> of the 30% solution.
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