Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Algebra - Equations - Simultaneous equations.
Test Yourself 1 - non-linear and multiple equations.


 

Solve the following non-linear equations simultaneously:

 

Line and a parabola.

y = x + 2

y = x2

Answer.x = -1 of x = 2.

 y = x + 3

y = 4x2

Answer.x = -0.75 or x = 1.

 y = x2 - 1

x - y + 1 = 0

Answer.x = -1 or x = 2.

y = x2 - x - 2

y = x + 6

 

    

 

2 parabolas.

y = x + x2 - 10

y = 3x2 - 2x - 19

 y = 4x2 + 41x +104

y = 3x2 + 37x + 104

 y = 2x2 + 6x + 91

y = (x + 8)2

 

 

 

Answer.a = 2 and r = 3.

    

 

Line and a circle.

x2 + y2 = 5

x + y = -1

 Find only the y values for the point of intersection of

x + 2y = 1

x2 + y2 = 4

Hint.Change the subject of the first equation to x = and then substitute into Equation 2.
As the factorisation is not possible, use the quadratic formula to solve for y (and leave as a surd). 

 (x + 1)2 + (y - 2)2 = 25

y = 3x

 

 

Line and an ellipse/hyperbola.

2x2 + y2 = 11

2x - y = -1

 xy = 8

y = x + 7
xy = 2

x - 2y = 3

 xy = -6

3x - 2y = 12

 x2 - 2y2 = 5

x - y = 1

Hint.With a coefficient of 1, it seems the substitution of the equation
x = into the second equation is the way to go.
There are however 2 substitutions to make for x and only 1 for y. So rewriting the first equation to y = (even having a fraction) is the better approach.

Other.

Answer.a = 2 and r = 3.

Answer.a = 10 and r = 0.5.

Answer.x = 2 and y = -3.
x2 - y2 = 7

x2 + y2 = 13

 

 x2 + y2 = 20

xy = 8

 

x + 2y = 5

2xy - x2 = 3