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...

8 months ago

Answered by Anthony, an A Level Maths tutor with MyTutor

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


£22 /hr

Molly B.

Degree: Mathematics (Bachelors) - Exeter University

Subjects offered: Maths, Biology


“About Me: I am a maths student at the University of Exeter and have a genuine love for my subject, which I hope my tutorials will instill in you too. I tutored Cores 3 and 4 of A level during my own exams, and helped my tutee to atta...”

£30 /hr

Chris S.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered: Maths, Spanish+ 1 more

Further Mathematics

“ I am extremely passionate about mathematics and I love the Spanish language! ”

£20 /hr

William D.

Degree: Mechanical Engineering (Masters) - Leeds University

Subjects offered: Maths, Physics+ 2 more


“I am a British student currently in the first year of a masters degree in Mechanical Engineering at the University of Leeds. Having previously lived in Spain and Italy, I am fluent in both languages and have excellent communications s...”

About the author

Anthony H.

Currently unavailable: for regular students

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

Differentiating (x^2)(sinx) Using the Product Rule

Find the equation of the tangent to the curve y = 2 ln(2e - x) at the point on the curve where x = e.

How do I know which method of diffirentiation to use?

Find the values of x, where 0 < x < 360, such that x solves the equation: 8(tan[x])^2 – 5(sec[x])^2 = 7 + 4sec[x]

View A Level Maths tutors


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