Question

5. simon and his sister are playing a guessing game. simon tells his sister that he is thinking of two positive numbers. the first number minus the second number is 15. the square of the first number minus 20 times the second number is equal to 300.

a. write a system of one linear and one quadratic equation to represent the two numbers. be sure to define your variables.

b. solve the system of equations.

c. what are the two numbers that simon is thinking of? explain your reasoning.

1. Thunguyet
a) The linear equation is x – y = 15 and
the quadratic equation is x^2 – 20y = 300.
b) The numbers are x = 0, y = -15 and x = 20, y = 5
c) The two numbers that Simon is thinking of is x = 20, y = 5.
In this question,
Let the first number be x and
The second number be y
Part a:
The first number minus the second number is 15, then the linear equation is
⇒ x – y = 15 —— (1)
The square of the first number minus 20 times the second number is equal to 300, then the quadratic equation is
⇒ x^2 – 20y = 300 ——- (2)
Part b:
Solve the system of equations
From (1) ⇒ y = x – 15 ——- (3)
Then, equation 2 becomes
⇒ x^2 – 20(x-15) = 300
⇒ x^2 – 20x + 300 = 300
⇒ x^2 – 20x = 300 – 300
⇒ x^2 – 20x = 0
⇒ x(x-20) = 0
⇒ x = 0 or x = 20
When x = 0, y = -15 and
When x = 20, y = 5
Part c:
The two numbers that Simon is thinking of is
The value of x and y are positive.
So, x = 20 and y = 5.
Hence we can conclude that
a) The linear equation is x – y = 15 and
the quadratic equation is x^2 – 20y = 300.
b) The values are x = 0, y = -15 and x = 20, y = 5
c) The two numbers that Simon is thinking of is x = 20, y = 5.