Just need to be certain

Liveit

Hi,

Right i'll just put this image in here first.




Just wondering about simultaneous equations. In the blue box he subtracts one equation from another but in the red box he adds them together. Both ways are ok I assume? And can you manipulate the equations any way you like before adding or subtracting them such as multiplying/dividing?

amen

The problem is that you have 3 variables x,y and z so you need to reduce the equations to two variables either x,y or x,z or y,z.

If you take the equation in the blue box and add them you end up with
4x+ 2z = 2 and in the red box you end up with 5x-2y = 9
which still leaves you with 3 variables of x,y,z

By subtracting those in the blue box the zs go away leaving you with just x and y so you can then get a value for x then y then z

Michael Collins

Yep, you can in general add, subtract or do whatever you want with 2 equations, provided you do the same on both sides of the equals sign.

Likewise with a single equation on its own - you can multiply across or divide or add or subtract (and more) once you do it to both sides of the equation. Think of it this way: an equation says one thing is equal to another thing; if you change both sides equally, this will still be true.

There are a few exceptions (such as division by zero), which are not allowed.

COUCH WARRIOR

adding and subtracting equations is one way of doing these. You can also do these by substitution.

The idea of this approach is to use two of the equations to express two of the variables say x and z in terms of y. Then using the third to solve for y

Equation (i) can be rewritten as x = -1/3(y+z) by subtracting (y+z) from both sides and dividing by 3

Substituting this into Equation (ii) for x gives -1/3(y+z)-y+z=2
multiplying across by 3 gives -4y+2z=6
or z=3+2y by dividing across by 2 and adding 2y to both sides

substituting back into our reformulated of Equation (1) gives x=-1/3(y+(3+2y))
or x=-y-1 by multiplying out the brackets

so we now have from equation (i) and (ii) x=-y-1 and z=3+2y, we can now substitute these into equation (iii) to get
2(-y-1)-3y-(3+2y)=9 or by multiplying out the brackets -7y-5=9 or by adding 5 to both sides -7y=14 or y=-2

(Note if you try to substitute your expressions of x and z in terms of y into either of the first two equations the y terms will cancel, if you've done it properly. :)I like this as an error check just over half way through the problem, which the method in the worked solution doesn't have:()

and as we have x=-y-1=-(-2)-1=1
and z=3+2(-2)=-1

I've not analysed if this method is quicker than the one in the worked example, but you can always do the same method without having to figure out whether to add or subtract the equations and it has the option to check after the half way point.
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These equations are designed so that the computations are relatively straight forward, I'm not sure if you get marks for method, but for me looking only at two of the variables say y and z, I can see that adding 2 times equation (i) to equation (iii) gives me 8x-y+z=9, which has the same y and z terms as equation (ii). So I simply add 2 times equation (i) plus equation (iii) minus equation (ii) giving me 7x=7 or x=1 and its straight forward after that. I wouldn't recommend this approach, you might spend more time staring at it than you'd take doing it out the long way.

Liveit

Thanks a lot people! It's all clear now