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May 20th, 2014, 06:05 PM   #1
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Simultaneous equations

solve this set of simultaneous equations.

7x - 4y = 37
6x + 3y = 51

thank you for all of the help given pre hand.

Last edited by skipjack; May 24th, 2014 at 08:32 PM.
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May 20th, 2014, 06:43 PM   #2
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Well you can start like this:

$$7x-4y=37\Rightarrow 21x-12y=111$$

$$6x+3y=51\Rightarrow 24x+12y=204$$

Then just add the equations and solve for x. Can you continue then solve for y?
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May 20th, 2014, 09:29 PM   #3
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Originally Posted by ron246 View Post
7x - 4y = 37
6x +3y = 51
Divide 2nd equation by 3: 2x + y = 17
So y = 17 - 2x
Substitute in 1st equation:
7x - 4(17 - 2x) = 37
Solve for x..then for y.

However, Eddie's way is usually the easiest...
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May 21st, 2014, 10:26 AM   #4
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Well, if you're not aware of the basics, then it becomes almost
impossible to conduct a classroom here...
Try here:
Two equations in two unknowns - Math Central

You can get other sites by googling "2 equations, 2 unknowns"

Come back if you have questions...
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May 21st, 2014, 02:20 PM   #5
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Originally Posted by ron246 View Post
Hi, I don't understand the answer. How do I continue to find x and y? Thank you
Type the keywords 'simultaneous equations' into the search box on Youtube and hit enter.

You should find plenty of videos that cover this subject matter. Those who seek shall find.

Last edited by skipjack; May 24th, 2014 at 08:35 PM.
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May 22nd, 2014, 02:33 AM   #6
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Math Focus: Physics, mathematical modelling, numerical and computational solutions
There are two ways to solve simultaneous equations:

Method 1: Elimination:

This method involves adding or taking away the equations in such a way so that the y or x is eliminated. Then you have an equation with just an x or just a y, so it can be solved.

This is the method that eddybob123 posted: the steps he conducted were as follows:

1. multiply the first equation by 3
2. multiply the second equation by 4
3. look at the y terms... one says $\displaystyle -12y$ and the other says $\displaystyle +12y$.

$\displaystyle 21x - 12y = 111$
$\displaystyle 24x+12y = 204$

4. add both equations together (you can separate out the x bits, y bits and numbers if you like, but I normally just write it out underneath), so

$\displaystyle 21x - 12y = 111$
$\displaystyle 24x+12y = 204$
$\displaystyle 45x + 0y = 315$

If the above is confusing, you might want to try to separate out the bits and see if it helps you, as I've done below. If not, don't bother with it.

y bit:
$\displaystyle -12y + +12y = -12y + 12y = 0$

so you're left with

x bit:
$\displaystyle 21x + 24x = 45x$

numbers bit:
$\displaystyle 111 + 204 = 315$


$\displaystyle 45x = 315$

Dividing both sides by 45 gives

$\displaystyle x = \frac{315}{45} = 7$

Now we know that x = 7, we substitute this back into any one of the equations we started with to get y.

Taking the second one (it looks nice! You can use the first one if you fancy.):

$\displaystyle 6x+3y = 51$
$\displaystyle 6\times7 +3y = 51$
$\displaystyle 42+3y = 51$
$\displaystyle 3y = 51 - 42 = 9$
$\displaystyle y = \frac{9}{3} = 3$

So the answer is x=7, y = 3.

You might be thinking "I added the equation above... why was that?" It's because the signs were different. My teacher always said "remember SSAD... subtract is same, add if different"

So... we added the equations above because the signs on the 12y were different (+ and -). If they are the same (+ and + or - and -), you subtract the two equations instead.

Method 2: Substitution.

Rearrange one equation for x or y, then substitute this into the second. This is the superior method for more difficult simultaneous equations.

Take first equation and rearrange for y (or x, if you fancy. I'm going to pick y though):

$\displaystyle 7x - 4y = 37$
$\displaystyle 7x = 37 + 4y$
$\displaystyle 4y = 7x - 37$
$\displaystyle y = \frac{7}{4}x - \frac{37}{4}$

then substitute this into the other equation.

$\displaystyle 6x + 3y = 51$
$\displaystyle 6x + 3\left(\frac{7}{4}x - \frac{37}{4}\right) = 51$
$\displaystyle 6x + \frac{21}{4}x - \frac{111}{4} = 51$
$\displaystyle \frac{24}{4}x + \frac{21}{4}x - \frac{111}{4} = \frac{204}{4}$
$\displaystyle \frac{45}{4}x = \frac{305}{4}$
$\displaystyle x = \frac{305}{4} \times \frac{4}{45} = \frac{305}{45} = 7$

Then resubstitute the answer for x back into the equation as in method 1 to get y (y=3).

As you can see, method 1 is much simpler, so for simultaneous equations like the ones you're getting, only use method 1 for your work at the moment. Don't bother with the other one, but remember that it exists if someone surprises you in the future with a crazy simultaneous equation
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