Question

​How many ordered pairs `(x,y)` satisfy the system of equations `x​=2y+5` and `y​=(2x−3)(x+9)`?

The choices are `0, 1, 2,` or infinitely many.

What is the quickest way to solve this without graphing? (It's for a standardized test practice.)

Answer 1

Ordered pairs are `2` which are `(1.414,-1.793)` and `(-8.664,-6.832)`

Explanation:

Number of ordered pairs ought to be `2` as `y=(2x-3)(x+9)` is a quadratic equation.

We can get them by ptting `x=2y+5` in above equation, which gives us

`y=(2(2y+5)-3)(2y+5+9)`

or `y=(4y+7)(2y+14)=8y^2+70y+98`

or `8y^2+69y+98=0`

As discriminant is `Delta=b^2-4ac=69^2-4xx8xx98=1625>0`, we have two ordered pairs and using quadratic formula, we get them as shown below.

`y=(-69+-sqrt(69^2-4xx8xx98))/16`

= `(-69+-sqrt(4761-3136))/16`

= `(-69+-sqrt1625)/16`

= `(-69+-40.3113)/16`

= `-1.793` or `-6.832`

And `x=2xx-1.793+5=1.414` or `x=2xx-6.832+5=-8.664`

Hence, ordered pairs are `(1.414,-1.793)` and `(-8.664,-6.832)`

graph{(y-(2x-3)(x+9))(x-2y-5)=0 [-20, 20, -10, 10]}