Algebra/Chapter 9/Quadratic Systems

Solving Systems of Simultaneous Equations Involving Equations Of Degree 2

The substitution method should be used for efficiency when solving nonlinear simultaneous equations, unless other methods such as the graphing method provide clear and simple solutions quickly (when they would be faster than substitution).

Example: Solve the system of simultaneous equations.

$y^2+(2x+3{)}^2=10$

$2x+y=1$

With the second equation, make a given term (here, 2x should be used) the subject.

$2x=1-y$

Substitute the third equation into the first, and through factorization of the resulting, simplified quadratic with one variable the solutions can be found.

$y^2+((1-y)+3{)}^2=10$

$y^2+(4-y{)}^2=10$

$2y^2-8y+16=10$

$y^2-4y+3=0$

$(y-1)(y-3)=0$

Hence we know $y=1$ or $y=3$

Then, we calculate that the two possibilities are: $y=1$, $x=0$ or; $y=3$, $x=-1$