Like many things in maths they look terrifying initially, but actually, when it clicks it will seem like common sense!

Simultaneous equations: they look terrifying, and to be honest, until you get your head around it, they are (and for me, that took quite a while…)  

4x+y=14

5x+2y=19  

Our first goal is to establish whether we are going to be adding or subtracting.

If the signs (the add or subtract) in the middle are the same, we subtract (Same Signs Subtract SSS)

If they are different, we add (Alternate Signs Add).  

The next thing we need to do is to get the number in front of the y the same.

To do this we need to multiply everything in the top line by 2, and everything in the bottom line by 1. 4x+y=14 (x2)

5x+2y=19 (x1)  

8x+2y =28

5x+2y=19

Now the value of the y’s are the same and we know that they are both add we can start to do the actual working out.

(We need to subtract downwards)  

8x-5x=3x

2y -2y is nothing which is why we want to make them the same because then life becomes so much easier

28-19 = 9

So now we have: 3x = 9

So, 3x something = 9.

That means that x must equal 3. 3x3=9.  

Now we know that x=3, we can substitute that back in:

Take your pick which one you want to substitute back into

4x+y=14 (4x3)+y=14  

12+y=14 12 + something (y) = 14

Therefore our ‘y’ must be 2.

To double check, pick another line: 5x+2y=19 (5x3)+(2x2) =19 15+4= 19  

We’re right 😊

Had the signs in the middle been different we would have gone through the same process but subtracted.  

Using the example:

3x+5y=19

4x-2y=-18  

This time the signs are different so ASA (alternate signs add).

Start by multiplying everything in the top line by 2 and everything in the bottom line by 5 to make the value of the y’s the same.

3x+5y=19 (x2)

4x-2y=-18 (x5)  

6x+10y = 38

20x -10y =-18

(Alternate signs add)  

6x – 20x = -14x

10y-10y = nothing so we can ignore those now

38- -18 = (subtract a negative = a positive) 38+18 =56

Giving us: -14x = 56 (56 divided by -14 = -4)

X= -4

(Sorry this example is harder than I anticipated)

But now we can substitute the -4 back in for the x

Pick a line, I’ve chosen:

3x+5y=19

(3x -4) +5y =19

-12 +5y =19 (add 12 to both sides)

5y = 31 Y = 6.2  

Pick another line and we can check our answer: 6x+10y = 38

(6x-4) + (10x6.2) =38

-24 +62 = 38  

Once again, it works.

I’ll do a video on those to make it easier to understand, if you want it and the link isn’t on the website nudge me and I will make sure I get it done and get it over to you instead of just adding it to my to do list.