6x+y^2=10 and 3x-y=5 find all solutions and determine whether or not the system is linear or nonlinear?

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Answer 1 · 2018-08-05T23:26:50

$(1,-2)$ and $(5/3,0)$
The system will be nonlinear because it includes a quadratic equation.

This is a system of equations, so solving one term one in terms of the other will allow you to find the solution

$-y=-3x+5$
$y=3x-5$

Then plug in the value of $y$ into the first equation

$6x+(3x-5)^2=10$

Solve for $x$

$6x+9x^2-30x+25=10$
$9x^2-24x+15=0$
$3(x^2-8x+5)=0$
$3(x-1)(x-5/3)=0$

$x=1, 5/3$

Now that we know the x coordinates of the solutions, we can plug them into one of the equations to solve for the y coordinates

$3(1)-y=5$
$y=-2$

$3(5/3)-y=5$
$y=0$