# Thread: Simultaneous Equation, modelling with algebra and algebraic proof.

1. ## Simultaneous Equation, modelling with algebra and algebraic proof.

1)I've spent ages trying to work this out but keep getting stuck by not being able to factorise the last bit, however this may be happening because I'm going wrong along the way.

2x^2+xy=-1
y-x=4

2)I keep getting the answer to be one and I can't figure out why.

If the numerator and demonimator of a fraction and both increased by 3 the fraction is equivalent to 2/3. If the numerator and denominator are both increased by 11 the fraction is equivalent to 4/5. What is the original fraction

3) I just don't have a clue and don't know where to start.

A small swimming pool can br filled with two pipes in 3 hours. If the larger pipe alone takes 8 hours less than the smaller pipe to fill the pool, find the time in which it will be filled by each pipe singly.

4)
Firstly, I don't really understand what 'base' in the question means and I cannot find the solution.

8^(n+1) = 2^n

5) This is the first time I have come across algebraic proof and this being some extra summer work I have not been taught how to prove it.

i) Prove that the sum of the squares of any two consecutive intergers is an odd number.

ii) Prove that the difference between the squares of two consecutive odd numbers is always devisible by 8.

Thanks to all of you who help me,

Max

2. 1.
$2x^2 + xy = -1$
$y - x = 4$.

Rearranging the second equation gives

$y = x + 4$

and substituting into the first gives

$2x^2 + x(x + 4) = -1$

$2x^2 + x^2 + 4x = -1$

$3x^2 + 4x + 1 = 0$

$3x^2 + 3x + x + 1 = 0$

$3x(x + 1) + 1(x + 1) = 0$

$(x + 1)(3x + 1) = 0$

$x + 1 = 0$ or $3x + 1 = 0$

$x = -1$ or $x = -\frac{1}{3}$.

Substituting back into $y = x + 4$ gives

$y = -1 + 4$ or $y = -\frac{1}{3} + 4$

$y = 3$ or $y = \frac{11}{3}$.

So $(x, y)= (-1, 3)$ and $(x, y) = \left(-\frac{1}{3}, \frac{11}{3}\right)$ are the solutions.

3. 5. a) You can write every odd integer as $2n + 1$ where $n$ is another integer.

So two consecutive odd numbers would be $2n + 1$ and $2n + 1 + 2$.

What do you get when you add them together?

b) Any odd integer can be written as $2n+ 1$ where $n$ is an integer.

So the two consecutive odd integers are $2n + 1$ and $2n + 1 + 2$.

Their squares are $(2n + 1)^2$ and $(2n + 3)^2$.

What is their difference?

4. Thanks a lot for the help that's cleared things up for those questions. Ill keep working on the others.

Thanks again,

Max