Math Help - Multiple Simultaneous Equations

1. Multiple Simultaneous Equations

How do I solve these by either substitution/ elimination? I'd also like to know how to solve them using matrices.

$U_n = an^{2}+bn+c=0$
$U_1=4$
$U_2=10$
$U_3=18$

I need to find the values of the constants a, b, c are.

2. Originally Posted by Viral
How do I solve these by either substitution/ elimination? I'd also like to know how to solve them using matrices.

$U_n = an^{2}+bn+c=0$
$U_1=4$
$U_2=10$
$U_3=18$

I need to find the values of the constants a, b, c are.
So it seems that $U_n$ in general is the result of a,b and c combined with the value of n for each equation. How was $U_1$ formed? The same a,b and c were used, just a different n. You have 3 unknowns and potentially 4 equations so that's all you need.

3. Yeah I got that, and I sorted the U1, 2 and 3 into their respective equations. I've just never been shown how to solve simultaneous equations with more than 2 equations.

4. Originally Posted by Viral
How do I solve these by either substitution/ elimination? I'd also like to know how to solve them using matrices.

$U_n = an^{2}+bn+c=0$
$U_1=4$
$U_2=10$
$U_3=18$

I need to find the values of the constants a, b, c are.
Are
$U_1=4$
$U_2=10$
$U_3=18$

constant fubctions of x?

5. Not sure what you mean.

$U_1 = 16a + 4b + c =0$

6. Originally Posted by Viral
Yeah I got that, and I sorted the U1, 2 and 3 into their respective equations. I've just never been shown how to solve simultaneous equations with more than 2 equations.
Gotcha. Well you can use substitution here but it's tricky and a lot of algebra. If you want to use matrices like you said, first write out three equations. Make sure they are all in the same form, meaning the a's, b's and c's line up. Then make a 3x3 matrix of the left hand side of these three equations. Make another matrix that is 3x1 matrix of the right hand side of the three equations. If the first one is A and the second one is B, then take $A^{-1}B$ and you'll get a 3x1 matrix as your answer which is the solution for a,b and c.

7. Hmm, actually we haven't learned how to work out the determinant of 3x3 matrices, only 2x2 matrices. How would I do this the algebraic way?

8. Originally Posted by Viral
Hmm, actually we haven't learned how to work out the determinant of 3x3 matrices, only 2x2 matrices. How would I do this the algebraic way?
Normally calculator's are allowed for this method.

Anyway, if you want to do it by hand, it's just like with two variables. Solve an equation for one variable in terms of the others. You can do this a couple times and get two equations with two unknowns. Solve them normally then use that to find the 3rd solution.

9. Unfortunately calculators aren't allowed. I've tried doing what you've suggested and I've ended up with like a page of working out :S . I'll try again in the morning.

10. Ok, I've tried forever and I just can't work it out =\ . I somehow got c = 2 which I don't think is correct.

11. Originally Posted by Viral
How do I solve these by either substitution/ elimination? I'd also like to know how to solve them using matrices.

$U_n = an^{2}+bn+c=0$ Mr F says: So ${\color{red}U_n = 0}$. Then how can you have non-zero values (below) for the cases n = 1, 2 and 3?
$U_1=4$
$U_2=10$
$U_3=18$

I need to find the values of the constants a, b, c are.
None of this makes any sense to me. See the red.

Do you mean the following?:

4 = a + b + c .... (1)

10 = 4a + 2b + c .... (2)

18 = 9a + 3b + c .... (3)

The following will give you two equation in a and b:

Equation (2) - equation (1):

Equation (3) - equation (1):

Solve them for a and b. Then use those values in one of the equations to solve for c.

12. Thanks, from that I got:

$\begin{array}{rcrcrc}a=1\\b=3\\c=0\end{array}$

Is that correct?

13. Originally Posted by Viral
Thanks, from that I got:

$\begin{array}{rcrcrc}a=1\\b=3\\c=0\end{array}$

Is that correct?
Do these answers work when you substitute them into the equations ....?