a The variables x, y, and z are the amounts in pounds of almonds, hazelnuts, and raisins, respectively.
B
b See solution.
C
c We need to buy 2.5 lb of almonds, 3.5 lb of hazelnuts, and 3 lb of raisins.
Practice makes perfect
a Notice that the coefficients for the variables in the second equation coincide with the price per weight of each of the snacks we want to include in the mix. The right-hand side equals the total amount of money to be spent.
You have $15 to buy almonds for $2.45per lb,
hazelnuts for $1.85 per lb, and raisins for $ 0.80
per lb.
& (I) 2.45x + 1.85y + 0.8z =15 & (II) & (III)
We can conclude that x, y, and z are the amounts in pounds of almonds, hazelnuts, and raisins, respectively. The first equation tells us that the sum of the total weight of the snacks is 9 lb, representing the first part of the exercise's wording.
Suppose you want to fill nine 1-lb tins with
a snack mix.
x+y+z = 9 & (I) 2.45x + 1.85y + 0.8z =15 & (II) & (III)
Finally, the third equation represents the proportions we want in our mix.
You want the mix to contain and equal amount
of almonds and hazelnuts and twice as much
of the nuts as the raisins by weight.
x+y+z = 9 & (I) 2.45x + 1.85y + 0.8z =15 & (II) x+y = 2z & (III)
b We will now solve the system we found in Part A. There is no unique correct way to do this. However, notice that we can isolate x or y straight away from Equation (III). We can then use the Substitution Method to reduce the system to one with only two equations and two variables.
x + y + z = 9 & (I) 2.45x +1.85y +0.8z = 15 & (II) x + y = 2z & (III) 1cm
Solve for $x$ from Equation (III) 1.3cm
x = 2z-y 3.2cm
Substitute $x$: 5cm
In Equation (I) 1.55cm In Equation (II) 0.5cm
2.45( 2z-y)+1.85 y 0.9cm 2z-y + y +z = 9 0.3cm
+0.8z=15 5.5cm
4.9z-2.45y +1.85 y 1.75cm 3z =9 1.3cm
+0.8z=15 5.5cm
5.7z-0.6y =15 4.5cm
We obtain a system of two equations 1.3cm
with two variables 2.6cm
5.7z-0.6y =15 & (I) 3z =9 & (II) 2.1cmWe can now proceed to solve this new system of two equations as usual. Notice that in Equation (II) we can solve for z in a few steps. We can then use the Substitution Method once a again.
The solution for the system is then x=2.5, y=3.5, and z=3.
c As we found in Part B, the solution for the system of equations is x=2.5, y=3.5, and z=3. In Part A we concluded that these variables represent the amount in pounds for the almonds, hazelnuts, and raisins, respectively. Hence, we need to buy 2.5 lb of almonds, 3.5 lb of hazelnuts, and 3 lb of raisins.
