The problem by modeling and solving a system of linear equations.

Procedure used:
"Model and solve system of linear equations
1) Identify the category of given problem.
2) Recognize and algebraically name the unknowns.
3) Translate the problem statement as a system of linear equations using the variables from Step 2.
S 4) Solve the system as obtained in Step 3.
5) State the answer statemen."
Calculation:
Step 1:
The given problem is a mixer problem where ${\displaystyle 60\mathrm{\%}}$ sugar solution and ${\displaystyle 30\mathrm{\%}}$ sugar solution are mixed together to make ${\displaystyle 51\mathrm{\%}}$ sugar solution where quantity of 30% sugar solution is to be calculated.
Step 2:
There are two unknowns in the problem. Let total amount of 60% sugar solution is a and that of 30% sugar solution is s.
Step 3:
The mixer problem is summarized is Table 1.
$\begin{array}{|ccc|}\hline Name& \text{Amount(in liters)}& Concentration\\ 60\mathrm{\%}sugarsolution& a& 60\mathrm{\%}\\ 30\mathrm{\%}sugarsolution& s& 30\mathrm{\%}\\ \hline\end{array}$
Total amount of ${\displaystyle 51\mathrm{\%}}$ sugar solution is 10 quarts.
${\displaystyle a+s=10}$
In the question it is given that ${\displaystyle 60\mathrm{\%}}$ sugar solution and ${\displaystyle 30\mathrm{\%}}$ sugar solution are mixed together to make 51% sugar solution.
${\displaystyle 0.60a+0.30s=0.51\left(10\right)}$
${\displaystyle 0.60a+0.30s=5.1}$
Thus, the system of linear equations is:
${\displaystyle a+s=10\left(1\right)}$
${\displaystyle 0.60a+0.30s=5.1\left(2\right)}$
Step 4:
Substitute the value of s from equation (1) to (2).
${\displaystyle 0.60a+0.30s=5.1}$
${\displaystyle 0.60a+0.30(10-a)=5.1}$
${\displaystyle 0.60a+3-0.30a=5.1}$
${\displaystyle 0.30a=5.1-3}$
Solve the equation above to obtain the value of variable p as:
${\displaystyle 0.30a=2.1}$
${\displaystyle a=\frac{2.1}{0.30}}$
${\displaystyle a=7}$
Substitute the value of the variable a in equation (1) to obtain the value of the variable s as follows:
${\displaystyle \left(7\right)+s=10}$
${\displaystyle s=10-7}$
${\displaystyle s=3}$
Thus, the values of unknowns defined is Step 2 are ${\displaystyle a=7}$ and ${\displaystyle s=3}$.
Step 5:
To check the solution obtained is Step 4, the total pounds are ${\displaystyle 7+3=10}$ and the amount of 51% sugar solution is 10 quarts ${\displaystyle 0.60\left(7\right)+0.30\left(3\right)=0.51\left(10\right)}$.
Step 6:
Hence, the amount of 30% sugar solution is 3 quarts.