When solving a simultaneous equation like this:

Question

When solving a simultaneous equation like this:

$2y - x = 4 $

$2x² + 3y² = x + 4y = 17 $

How do you express this second equation? I know how to solve simultaneous equations. I'm not just sure of how to express it.

Attempt: I solved it in two ways. I made it in this form:

$2x² + 3y² - x - 4y = 17 $

And then solved it. It became a quadratic equation and I got two values for x and y which when I replaced gave me the equation back. I'm not just sure that's how to express it.

I also solved it by separating the second equation into two different equations

$2x² + 3y² = 17 $

$x + 4y = 17 $

Which when I solved didn't equate when I replaced it.

So how is it supposed to be? Thanks

Answer 1

You have in fact three equations and two unknowns, so there is a possibility that the system has no solution. To find out, we choose the two easier (independent) equations and check the solution in the third. $$ \cases{ 2y-x=4,\\ 4y+x=17,} $$ implying that $6y=21$, so $y=7/2$ and $x=3$. To check the third equation: $$2\cdot3^2+3(7/2)^2=\frac{219}4\neq17$$ So the system has no solution.

Answer 2

From $2x^2+3y^2=x+4y=17$ I deduce that $2x^2+3y^2=17$ as well as $x+4y=17$ You can simply graph to see that the ellipse and the line do not intersect...