# Methods for solving simultaneous equations

1. Jun 18, 2009

### TheAkuma

Hey everyone. Okay, first of all I'm not sure if these equations are simultaneous or linear so please correct me if I'm wrong. I need to know a simple procedure to tackle on these equations.Thanks.

a) y=x-6 and y=2x-14
b) y=0.5x-7 and y=3x-22
c) 9y-4x+6=0 and 4x-2y-7=0

2. Jun 18, 2009

### CompuChip

Well, the first two are very easy because you have just "y" on the left hand side. Since obviously, y cannot have two different values at the same time, you must have y = y and therefore in the first case also x - 6 = 2x - 14 and in the second case 0.5x - 7 = 3x - 22. From this you can solve x in both cases, and then you just plug that value back in to also find y.

The third case is a little more tricky. I suggest rewriting one of the equations to isolate either x or y, then plug that into the other equation so you get a single equation in a single variable. For example, you can rewrite the second one to 4x = .... and then replace the 4x in the first equation by that expression (watch out for the minus sign in front!)

3. Jun 18, 2009

### Staff: Mentor

Alternatively - note, that on the left hand side of both equation you have 4x and -4x. If you add these equations side by side, x will cancel out.

Quite often this can be done just by multiplying one of the equations by a constant. For example

x + 2y = 7
2x + 3y = 6

multiply first equation by -2:

-2x - 4y = -14

-2x - 4y = -14
2x + 3y = 6
----------------
-y = -8

You may as well multiply second equation by 2 and then subtract it from the first equation, it will give the same effect.

4. Jun 18, 2009

### TheAkuma

Thank you so very much. That third one really made me scratching my head. Are you able to give me the order of operation to rewrite the 4x= please?

5. Jun 18, 2009

### TheAkuma

THanks. So you're saying all i have to do is just add the equations for c) together?

6. Jun 18, 2009

### Staff: Mentor

Try it.

7. Jun 18, 2009

### TheAkuma

I don't know what i keep doing wrong. I keep getting 7y=1

8. Jun 18, 2009

### TheAkuma

Oi dude, my Maths B exam is tomorrow and i need to know this question please

9. Jun 18, 2009

### TheAkuma

its alright, ii finally got it. Thanks

10. Jun 18, 2009

### Staff: Mentor

Good

I was AFK, but I see you managed to get it right on your own.

11. Jun 18, 2009

### HallsofIvy

For (c), the simplest way is to notice that the two equations, 9y-4x+6=0 and 4x-2y-7=0 Have "4x" and "-4x". If you add the two equations, they cancel leaving (9-2)y+ (6-7)= 0 or 7y- 1= 0. Then 7y= 1 so y= 1/7. Replace y in either of the original equations to get a single equation to solve for x.

By the way, the equations are both "simultaneous" (because there is more than one equation) and "linear" (because you have no powers of x and y except $x^1= 1$ and $y^1= y$ or more complicated functions of x and y.