Respuesta :
According to the information given in the exercise:
- The laboratory technician needs to make a 54 -liter batch of a 20 % acid solution.
- The combination has a batch of an acid solution that is pure acid with another solution that is 10% acid solution.
Let be "p" the amount of acid solution that is pure acid (in liters) and "t" the amount of solution that is 10% acid.
Using the information given, you can set up the following:
[tex]\begin{cases}p+t=54 \\ p+0.1t=(0.2)(54)\end{cases}[/tex]Notice that when you simplify the second equation:
[tex]\begin{cases}p+t=54 \\ p+0.1t=10.8\end{cases}[/tex]As you can see, the second equation represents the addition of the amount of 100% acid solution and the amount of 10% acid solution, which will result in a 20% acid solution.
Remember that a percent can be written as a Decimal Number by dividing it by 100:
[tex]\begin{gathered} \frac{100}{100}=1 \\ \\ \frac{10}{100}=0.1 \\ \\ \frac{20}{100}=0.2 \end{gathered}[/tex]Those are the coefficients used in the second equation.
To solve the system of equations, you can follow these steps:
1. Multiply the second equation by -1:
[tex]\begin{cases}p+t=54 \\ -p-0.1t=-10.8\end{cases}[/tex]2. Add the equations.
3. Solve for "t".
Then:
[tex]\begin{gathered} \begin{cases}p+t=54 \\ -p-0.1t=-10.8\end{cases} \\ ------------ \\ 0+0.9t=43.2 \\ \\ t=\frac{43.2}{0.9} \\ \\ t=48 \end{gathered}[/tex]4. Substitute the value of "t" into any original equation:
[tex]\begin{gathered} p+t=54 \\ p+(48)=54 \end{gathered}[/tex]5. Solve for "p":
[tex]\begin{gathered} p=54-48 \\ p=6 \end{gathered}[/tex]Therefore, you can determine that the answer is:
The laboratory technician can combine 6 liters of pure acid solution and 48 liters of 10% acid solution to get the desired concentration.
