# simultaneous equations

• Jan 7th 2012, 02:25 PM
David Green
simultaneous equations
I am trying to solve the simultaneous equations by algebraic method, I know the solutions for "a" and "b" are 0.46 and -5.2, I have used a few different methods and the latest is close but not correct.

Using elimination method I came up with;

12a + 3b = - 10
3a - 9b = 17

36a + 9b = - 30

9a - 9b = 51

36 - ( - 9)a + (9 - 9)b = -30 + 51

41a = 21

a = 0.512

3a - 3b = 17

3(0.512) - 3b = 17

6.15 - 3b = 17

3b = 17 - 6.15

b = -5.38

12(0.512) + 3(-5.38) = -10

3(0.512) - (-16.14) = 17.69

As can be seen by the solutions they are not a million miles out, but somewhere I am making an error?
• Jan 7th 2012, 02:35 PM
Quacky
Re: simultaneous equations
This is very promising work! With the first situation, you start off by correctly multiplying the first equation by three, which leads to the following equations:

$3a - 9b = 17$
$36a + 9b = - 30$

You then get confused. You multiply the first equation by $3$, which isn't necessary, and forget to multiply the $9b$ by $3$. Instead, if at this stage you simply add the equations, you get that $39a=-13$. This leads to $a=\frac{-13}{39}=\frac{-1}{3}$, which is not the answer you've been presented with. Are you sure you've written the question correctly?
• Jan 7th 2012, 02:50 PM
David Green
Re: simultaneous equations
Yes I have made an error as you say with the multiplication, but the equations (Original) are correct, I will try again?

Tried again a different method but can't get closer to the right solution at the moment?
• Jan 7th 2012, 03:50 PM
Quacky
Re: simultaneous equations
If the original equations are:

$12a + 3b = - 10$
$3a - 9b = 17$

...then the given answer is incorrect.
• Jan 7th 2012, 04:31 PM
HallsofIvy
Re: simultaneous equations
Quote:

Originally Posted by David Green
I am trying to solve the simultaneous equations by algebraic method, I know the solutions for "a" and "b" are 0.46 and -5.2, I have used a few different methods and the latest is close but not correct.

Using elimination method I came up with;

12a + 3b = - 10

12(0.46)+ 3(-5.2)= 5.52- 15.6= -10.08, not -10

Quote:

3a - 9b = 17
3(0.46)- 9(-5.2)= 1.38+ 46.8= 48.18, not 17.

The solutions you say you "know" are incorrect.

Quote:

36a + 9b = - 30

9a - 9b = 51

36 - ( - 9)a + (9 - 9)b = -30 + 51

41a = 21

a = 0.512

3a - 3b = 17

3(0.512) - 3b = 17

6.15 - 3b = 17

3b = 17 - 6.15

b = -5.38

12(0.512) + 3(-5.38) = -10

3(0.512) - (-16.14) = 17.69

As can be seen by the solutions they are not a million miles out, but somewhere I am making an error?
• Jan 8th 2012, 05:42 AM
David Green
Re: simultaneous equations
OK I have had another attemp, so far I can only get one of the two equations to work?

The previous thread points out that the solutions I have are incorrect, these solutions were calculated using a software program for simutaneous equations, therefore if they are indeed incorrect, then the software or a mathematical technique (Idea) is different to solving these equations?

12a + 3b = -10
3a - 9b = 17

I have multiplied out the first equation;

36a + 9b = -30

next I have subtracted the second equation;

36a + 9b = -30
3a - 9b = 17

a = 1.42

Using equation (1) I have worked out the value for b;

12(1.42) + 3b = -10
17 + 3b = -10
3b = -10 - 17

b = -9

substitute this value into equation (1)

12(1.42) + 3(-9) = 17 - 27 = -10

Up to here the values for a and b work, however I am either doing something wrong, or there is another mathematical method for solving these equations that I am not aware of, the values will not work for the second equation, and I am struggling to find another method?

any help now would be much appreciated.

P.S.

I have read in a Algebra book that the laws of basic arithmetic + and - do not equal - ?

Is this true or can I take it that the book has a typo error?

Thanks
• Jan 8th 2012, 08:28 AM
Quacky
Re: simultaneous equations
Remember that as I showed in post#2, your given answer is wrong. You are correct up to:

$36a + 9b = -30$
$3a - 9b = 17$

But if you subtract the equations, you get this:

$36a+9b-(3a-9b)=-30-17$
$33a+18b=-47$ ...which doesn't help us.

Rule: If the terms you are trying to eliminate have the same sign, subtract the equations. If the terms you are trying to eliminate have opposing signs, add the equations.

Here, the $9b$ is positive in $36a + 9b = -30$ but it's negative in $3a - 9b = 17$. This means, using previously stated rule, we add the equations.
• Jan 8th 2012, 10:05 AM
David Green
Re: simultaneous equations
following on from what you said in the previous thread, the equations;

12a + 3b = -10
3a - 9b = 17

36a + 9b = -30
3a - 9b = 17

This is where the confusion starts;

-30 + 17 = -13
+ 9 - 9 = 0
36 + 3 = 39

a = -0.33

b = -1.99

= -21.87, which is of course completely wrong.

The arithmetic I think is were I am getting this wrong.

I was always told that the following rules to arithmetic applied;

+ and + = +
- and - = +
+ and - = -

here in the equations I am now unsure, no matter which way I try to solve this I get one answer correct and one incorrect as follows;

12a + 3b = -30
3a - 9b = 17

36a + 9b = -30
3a - 9b = 17

33a = -47

a = -1.42

12(-1.42) + 3b = -10

-17 + 3b = -10

3b = -10 + 17

b = 2.33

12(-1.42) + 3(2.33) = -17 + 6.99 = -10

The next equation gives a solution completely wrong at -16.71 using those same values, so at the moment I am either way off track or these equations cannot be solved using integers?
• Jan 8th 2012, 10:21 AM
Quacky
Re: simultaneous equations
When you said that

$a = -0.33$
$b = -1.99$, you had the correct solution but for a rounding error. Remember where appropriate to leave your answers in an exact form. $a=\frac{-1}{3}$, which gives $b=-2$, which is the correct solution when we back substitute:

$12(\frac{-1}{3}) + 3(-2)$

$=\frac{-12}{3}-6$

$=-4-6$

$=-10$, as expected.