Please explain with complete step by step instructions.
#8)One type of lawn fertilizer consists of a mixture of nitrogen, phosphorus and potassium. An 80lb sample contains 8 move lbs of nitrogen and phosphorus than potassium. There is 16lb more potasssium than 5 times the amount of phosphorus. Let x be the amount of nitrogen, let y be the amount of phosphorus and let z be the amount potassium in the sample. Find the amount of each of the chemicals in the sample.
X = _____
y = _____
z = _____
(Hint: you must set up and solve a 3 by 3 systems of equations.)
To solve this problem, we need to set up a system of equations using the given information and then solve for the unknowns, x, y, and z.
Step 1: Assign variables to the unknowns. Let x represent the amount of nitrogen, y represent the amount of phosphorus, and z represent the amount of potassium in the sample.
Step 2: Write down the given information as equations.
From the problem, we have the following information:
- The total weight of the sample is 80 lbs: x + y + z = 80
- The sample contains 8 more pounds of nitrogen and phosphorus compared to potassium: x + y = z + 8
- There is 16 lbs more potassium than 5 times the amount of phosphorus: z = 5y + 16
Step 3: Solve the system of equations using substitution or elimination.
By substituting the third equation into the second equation, we get:
x + y = 5y + 16 + 8
Simplifying the equation:
x - 4y = 24 (Equation 4)
Now, we have two equations with two variables:
x + y + z = 80 (Equation 1)
x - 4y = 24 (Equation 4)
We can solve this system of equations by substitution or elimination. For simplicity, let's use the substitution method.
From Equation 4, we have:
x = 4y + 24
By substituting this value of x into Equation 1, we get:
(4y + 24) + y + z = 80
Combining like terms:
5y + z = 56 (Equation 5)
Now, we have two equations:
5y + z = 56 (Equation 5)
x - 4y = 24 (Equation 4)
To eliminate z, we can subtract Equation 5 from Equation 1:
(x + y + z) - (5y + z) = 80 - 56
Simplifying the equation:
x - 4y = 24 (Equation 4)
Now, we have:
x - 4y = 24 (Equation 6)
We can solve Equation 6 for x:
x = 4y + 24 (Equation 7)
Step 4: Substitute the value of x from Equation 7 into Equation 5 to solve for y.
5y + z = 56
4y + 24 - 4y + z = 56 (substituting x = 4y + 24)
Simplifying the equation:
z = 32
Step 5: Substitute the value of z into Equation 5 to solve for y.
5y + 32 = 56
5y = 24
y = 4.8
Step 6: Substitute the value of y into Equation 7 to solve for x.
x = 4(4.8) + 24
x = 43.2
Step 7: Verify the solution by checking all three equations with the obtained values of x, y, and z.
x + y + z = 43.2 + 4.8 + 32 = 80
x - 4y = 43.2 - 4(4.8) = 24
5y + z = 5(4.8) + 32 = 56
The obtained values of x = 43.2, y = 4.8, and z = 32 satisfy all three equations.
Therefore, the amount of each chemical in the sample is as follows:
X = 43.2 lbs
y = 4.8 lbs
z = 32 lbs