Linear-Quadratic systems question?

Question

Consider the system of equations. 

y = 4x^2 - 9x + 20 
y = 15x + k 

There are two values of x which are solutions to the system, one in which is three times as large as the other. 

Determine the value of k and hence the solution to the system.

Answer 1

The system is y = 4x ² - 9x + 20 and y = 15x + k

Equate the the y  forms.

4x ² - 9x + 20 =  15x + k

4x ² - 9x + 20 - 15x - k = 0

4x ² - 24x  + 20 - k = 0

4x ² - 24x  + (20 - k ) = 0

Solve the equation by using Quadratic formula.

Compare it to

Solutions are

According to given data, in the solutions one which is three times as large as the other.

So

Substitute the k  value in solutions.

Comments

Contd...

Substitute the x  and k  values in in y  = 15x  + k .

y  = 15(4.5) - 7

y  = 67.5 - 7

y  = 60.5

y  = 15(1.5) - 7

y  = 22.5 - 7

y  = 15.5

Solutions to the system (x , y ) = (4.5, 60.5) and (x, y ) = (1.5, 15.5).