To solve a mixture application involving solid ingredients, you can translate the problem's constraints into a system of equations by organizing the known and unknown amounts and values. For instance, suppose Carson wants to make $20$ pounds of trail mix using nuts and chocolate chips, with a target cost of $\$7.60$ per pound. If nuts cost $\$9.00$ per pound and chocolate chips cost $\$2.00$ per pound, we can determine the required amounts.

First, define the variables: Let $n$ represent the pounds of nuts and $c$ represent the pounds of chocolate chips.

Next, set up the system of equations. The first equation represents the total amount: $n + c = 20$. The second equation represents the total value. The value of the nuts is $9n$ and the value of the chocolate chips is $2c$. The total expected value of the mixture is $20(7.60) = 152$. Therefore, the second equation is $9n + 2c = 152$.

Finally, solve the system. Using the elimination method, multiply the first equation by $-2$ to get $-2n - 2c = -40$. Adding this to the second equation eliminates $c$, resulting in $7n = 112$, which simplifies to $n = 16$. Substituting $n = 16$ back into the first equation ($16 + c = 20$) yields $c = 4$. Thus, the mixture should consist of $16$ pounds of nuts and $4$ pounds of chocolate chips.