Question

A gardener has two fertilisers that contain different concentrations of nitrogen. One is 3% nitrogen and the other is 11% nitrogen. How many pounds of each should she mix to obtain 20 pounds of a 9% concentration?

1. Define the variables if they are not already defined in the problem, i.e. Let x be and y be ......... (10 marks).

2. How many equations do we need to model the problem? Write down the equations in terms of the variables (20 marks)

3. Solve the system of equations by substitution or elimination. Write down the steps clearly (20 marks).

4. Sketch the graph of this system of equations. Indicate the x-intercept and y-intercept of each equation. Also indicate the intersection point (40 marks).

5. Indicate the solution and make a recommendation to the gardener(10 marks).​

Answer 1

We want to see how many pounds of each fertilizer the gardener needs to mix.

1) First, we define the variables, these are:

• x = pounds of the 3% nitrogen concentration fertilizer.
• y = pounds of the 11% nitrogen concentration fertilizer.

2) Now we write the equations we need, one is:

x + y = 20lb

the other comes for the desired concentration:

(x*0.03 + y*0.11) = (x + y)*0.09

So we need to solve two equations.

3) The system is:

x + y = 20lb

(x*0.03 + y*0.11) = (x + y)*0.09

First we can replace (x + y) = 20lb in the second equation to get

(x*0.03 + y*0.11) =20lb*0.09

We also can write:

x = 20lb - y

Then replace that to get:

(20lb - y)*0.03 + y*0.11 = 20lb*0.09

y*0.08 = -20lb*0.03 + 20lb*0.09

y = 20lb*0.06/0.08 = 15lb

Then x = 20lb - y = 20lb - 15lb = 5lb

5) From this we can conclude that he must use 5lb of the 3% fertilizier and 15lb of the 11% fertilizer.

4) The graph can be seen below.

If you want to read more: