document.write( "Question 466772: A dehydrated patient needs a 3.24% saline IV. Unfortunately, the hospital only has bags of 1% and 9% saline solutions. How many liters of each of these solutions should be mixed together to yield 5 liters of the desired concentration? Please show all your work. \n" ); document.write( "
Algebra.Com's Answer #320049 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
x = number of liters of 1% solution.
\n" ); document.write( "y = number of liters of 9% solution.
\n" ); document.write( "you want a total of 5 liters of 3.24% solution.
\n" ); document.write( "your first equation is:
\n" ); document.write( "x + y = 5
\n" ); document.write( "this tells you that the total amount of solution will be 5 liters.
\n" ); document.write( "your second equation is:
\n" ); document.write( ".01*x + .09*y = .0324*5
\n" ); document.write( "this tells you that the mix of x liters of 1% solution + y liters of 9% solution will equal 5 liters of 3.24% solution.
\n" ); document.write( "you need to solve these 2 equations simultaneously to get your answer.
\n" ); document.write( "the equations are:
\n" ); document.write( "x + y = 5
\n" ); document.write( ".01*x + .09*y = .0324*5
\n" ); document.write( "simplify the second equation to get:
\n" ); document.write( ".01*x + .09*y = .162
\n" ); document.write( "your 2 equations are now:
\n" ); document.write( "x + y = 5
\n" ); document.write( ".01*x + .09*y = .162
\n" ); document.write( "solve for x or y in the first equation and then substitute in the second equation.
\n" ); document.write( "from the first equation, solving for y gets:
\n" ); document.write( "y = 5 - x
\n" ); document.write( "substituting for y in the second equation gets:
\n" ); document.write( ".01*x + .09*(5-x) = .162
\n" ); document.write( "since you have reduced the second equation from 2 unknowns to one unknown by applying the substitution from the first equation, you can solve for x in the second equation.
\n" ); document.write( "simplify the second equation to get:
\n" ); document.write( ".01*x + .09*5 - .09*x = .162
\n" ); document.write( "combine like terms and simplify further to get:
\n" ); document.write( "-.08*x + .45 = .162
\n" ); document.write( "subtract .45 from both sides of this equation to get:
\n" ); document.write( "-.08*x = .162 - .45
\n" ); document.write( "simplify this to get:
\n" ); document.write( "-.08*x = -.288
\n" ); document.write( "divide both sides of this equation by -.08 to get:
\n" ); document.write( "x = -.288 / -.08
\n" ); document.write( "simplify this to get:
\n" ); document.write( "x = 3.6
\n" ); document.write( "use your first equation to derive that y will be equal to 1.4
\n" ); document.write( "you have:
\n" ); document.write( "x = 3.6 liters
\n" ); document.write( "y = 1.4 liters
\n" ); document.write( "x + y = 5 becomes 3.6 + 1.4 = 5 which becomes 5 = 5, confirming the values for x and y solve the first equation.
\n" ); document.write( "substituting in your second equation of:
\n" ); document.write( ".01*x + .09*y = .162 gets you:
\n" ); document.write( ".01*3.6 + .09*1.4 = .06 which becomes .036 + .126 = .162 which becomes .162 = .162, confirming the values for x and y solve the second equation.
\n" ); document.write( "the values of 3.6 for x and 1.4 for y have solved both equations simultaneously, so they're good.
\n" ); document.write( "your answer is:
\n" ); document.write( "you need 3.6 liters of the 1% solution and 1.4 liters of the 9% solution to make 5 liters of 3.24% solution.
\n" ); document.write( " \n" ); document.write( "
\n" );