Question

If a nurse has 20% alcohol solution and 70% alcohol solution, but needs 15 liters of a 40% alcohol solution, how do you think the nurse should proceed? Let’s use a system of linear equations and two different ways of solving this system of equations.

First, we need to identify our variables, since we need to find out how much 20% alcohol solution (let’s call it Mixture A) and how much 70% alcohol solution (let’s call it Mixture B) we need.

Since our final mixture is a total of 15 liters, our first equation is A + B = 15.
Since our final mixture should have 40% alcohol solution, we need 6 liters of alcohol (.40*15) our second equation is 0.20A + 0.70B = 6.
Your job: Solve this system of equations for variables A and B by (1) graphing and by (2) either the substitution or substitution.

Answer 1

By solving the system of equations using both methods we get ,

Both graph intersects at (9,6). So A=9 and B=6

By substitution method we got A=9 and B=6

Given :

If a nurse has 20% alcohol solution and 70% alcohol solution, but needs 15 liters of a 40% alcohol solution

We are already given with 2 equations

$A + B = 15\\0.20A + 0.70B = 6$

Lets graph both the equations

While graphing we use variables   x  and y

Lets take A as x  and B as y

$x + y = 15\\0.20x + 0.70y = 6$

Lets graph first equation

$x+y=15\\y=-x+15\\slope =-1\\y intercept = 15$

Graph the above equation using slope and y intercept

Now we graph the second equation

$0.20x + 0.70y = 6\\+ 0.70y = -0.20x+6\\y=-\frac{2}{7} x+\frac{6}{0.7} \\slope =-\frac{2}{7} \\y intercept is \frac{6}{0.7}$

Graph is attached below

Both the lines intersects at (9,6)

A is 9  and B is 6

Now we solve using substitution method

$0.2A+0.7B=6\\2A+7B=60\\2A=60-7B\\A=\frac{60-7B}{2}$

Substitute it in first equation

$\frac{60-7B}{2}+B=15\\\frac{2\left(60-5B\right)}{2}=15\cdot \:2\\60-5B=30\\-5B=-30\\B=6$

Find out A by replacing B with 6

$A=\frac{60-7\cdot \:6}{2}\\A=9$

So, A is 9  and B is 6 using substitution method

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