AlgebraP1^
1
Non-linear simultaneous equations
The simultaneous equations in the examples so far have all been linear, that
is their graphs have been straight lines. A linear equation in, say, x and y
contains only terms in x and y and a constant term. So 7x + 2 y = 11 is linear
but 7x^2 + 2 y = 11 is not linear, since it contains a term in x^2.
You can solve a pair of simultaneous equations, one of which is linear and the
other not, using the substitution method. This is shown in the next example.EXAMPLE 1.36 Solve x + 2 y = (^7) ^1
x^2 + y^2 = (^10) ^2
SOLUTION
Rearranging equation ^1 gives x = 7 − 2 y.
Substituting for x in equation ^2 :
(7 − 2 y)^2 + y^2 = 10
Multiplying out the (7 − 2 y) × (7 − 2 y)
gives 49 − 14 y − 14 y + 4 y^2 = 49 − 28 y + 4 y^2 ,
so the equation is
49 − 28 y + 4 y^2 + y^2 = 10.
This is rearranged to give
5 y^2 − 28 y + 39 = 0
⇒ 5 y^2 − 15 y − 13 y + 39 = 0
⇒ 5 y(y − 3) − 13(y − 3) = 0
⇒ (5y − 13)(y − 3) = 0
Either 5 y − 13 = 0 ⇒ y = 2.6
Or y − 3 = 0 ⇒ y = 3
Substituting in equation ^1 , x + 2 y = 7:
y = 2.6 ⇒ x = 1.8
y = 3 ⇒ x = 1
The solution is either x = 1.8, y = 2.6 or x = 1, y = 3.
! Always substitute into the linear equation. Substituting in the quadratic will give
you extra answers which are not correct.
A quadratic in y which you
can now solve using
factorisation or the formula.