This sections looks at applications of systems of equations more closely.
To solve this system by substitution, notice that it will be easier to solve for one of the variables in the first equation. We’ll solve that equation for :
Now we will substitute for in the second equation:
Lastly, we can determine the value of by using the earlier equation where we isolated :
In summary, Notah charged to the Visa and to the Mastercard. We should check that these numbers work as solutions to our original system and that they make sense in context. (For instance, if one of these numbers were negative, or was something small like , they wouldn’t make sense as credit card debt.)
Mixture Problems
The next two examples are called mixture problems, because they involve mixing two quantities together to form a combination and we want to find out how much of each quantity to mix.
One quantity given to us in the problem is mL. Since this is the total volume of the mixed drink, we must have: Or without units:
To build the second equation, we have to think about the alcohol concentrations for the scotch, vermouth, and Rob Roy. It can be tricky to think about percentages like these correctly. One strategy is to focus on the amount (in mL) of alcohol being mixed. If we have milliliters of scotch that is alcohol, then is the actual amount (in mL) of alcohol in that scotch. Similarly, is the amount of alcohol in the vermouth. And the final cocktail is mL of liquid that is alcohol, so it has milliliters of alcohol. All this means: Or without units:
So our system is:
To solve this system, we’ll solve for in the first equation: And then substitute in the second equation with :
As a last step, we will determine using the equation where we had isolated :
In summary, LaVonda needs to combine mL of scotch with mL of vermouth to create mL of Rob Roy that is alcohol by volume.
As a check for the previous example, we can use estimation to see that our solution is reasonable. Since LaVonda is making a solution, she would need to use more of the concentration than the concentration, because is closer to than to . This agrees with our answer because we found that she needed mL of the solution and mL of the solution. This is an added check that we have found reasonable answers.
To set up our system of equations we define variables, letting be the amount of Columbian coffee beans (in pounds) and be the amount of Honduran coffee beans (in pounds).
The equations in our system will come from the total amount of beans and the total cost. The equation for the total amount of beans can be written as: Or without units: To build the second equation, we have to think about the cost of all these beans. If we have pounds of Columbian beans that cost per pound, then is the cost of those beans in dollars. Similarly, is the cost of the Honduran beans. And the total cost is for pounds of beans priced at per pound, totaling dollars. All this means: Or without units and carrying out the multiplication on the right: Now our system is:
To solve the system, we’ll solve the first equation for :
Next, we’ll substitute in the second equation with :
Since , we can conclude that .
In summary, Desi needs to mix pounds of the Honduran coffee beans with pounds of the Columbian coffee beans to create this blend. Our estimate at the beginning was pretty close, so we feel this answer is reasonable.