Solve the simultaneous equations y = x + 3, y^2 - x^2 + 3 = -6x

Here, we have two equations involving two variables, x and y. 

The question asks us to 'solve' the set of simultaneous equations.

Equivalently, it is asking us to find the x values (and corresponding y values) which satisfy both of the equations.

We can approach this by substituion:

We will rearrange one equation so that one of the variables, for example y, is the subject of the equation (i.e it is on one side of the equation with only x terms on the other).

Looking at the two equations, the first one is already in this form.

So now we can substitute y in terms of x into the second equation, giving:

(x+3)2 - x2 + 3 = -6x 

Now we have an equation in terms of one variable, x, which we should be able to solve.

Expand out the terms and group like terms so that we have the equation in a form that we know we can solve from:


x2 + 3x + 3x + 9 - x2 + 3 = -6x

Collecting like terms and simplyfying (x, x2 and constants) on the Left:

6x + 12  = -6x

Bringing all terms to the LHS:

12x + 12 = 0

Factorising out the 12:

12(x + 1) = 0

Now, x + 1 = 0 as 12 doesn't equal zero

So, rearranging:

x = - 1

Substituing into the simpler equation for y (y = x + 3) gives:

y = - 1 + 3 = 2

We have now finished as we have the x and the corresponding y value that satisfy the simultaneous equations.

Anthony H. 11 Plus Maths tutor, 13 plus  Maths tutor, GCSE Maths tuto...

1 year ago

Answered by Anthony, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£24 /hr

Ayusha A.

Degree: BEng electrical and electronics engineering (Bachelors) - Newcastle University

Subjects offered:Maths, Physics+ 1 more

Further Mathematics

“About me: I am a final year Electrical and Electronic Engineering student at Newcastle University. I took Mathematics, Further Mathematics, Chemistry and Physics as my A-level subjects. I did peer mentoring in university and also have...”

£20 /hr

Dan W.

Degree: Economics and Accounting (Bachelors) - Bristol University

Subjects offered:Maths, Economics


“I achieved top grades whilst juggling cricket at a high level. I’ve tutored for Young Einstein Tuition & been a Peer Mentor to those facing personal issues”

£24 /hr

Sam F.

Degree: Economics with Placement (Bachelors) - Bath University

Subjects offered:Maths, Physics+ 2 more

Extended Project Qualification

“Studying for BSc Hons Economics, A level economics, maths and physics. Able to tutor GCSE/AS/A2 Economics, Maths and GCSE physics! ”

About the author

£20 /hr

Anthony H.

Degree: Mathematics and Physics (BSc MMathPhys) (Masters) - Warwick University

Subjects offered:Maths, Physics+ 1 more


“About Me: I study Maths and Physics at University of Warwick. I have always had a passion for understanding how and why the world behaves as it does, and so I particularly love phyiscs and the maths behind it all. I am very patient a...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

The equation of a curve is x(y^2)=x^2 +1 . Using the differential, find the coordinates of the stationary point of the curve.

State the trigonometric identities for sin2x, cos2x and tan2x

Express (3+ i)(1 + 2i) as a complex number in the form a+bi where a and b are real numbers.

Differentiate f(x) = 2xlnx.

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss