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Math Help ( Simultaneous Equations and Linear Equations)

Can someone here explain how to do Simultaneous Equations and Linear Equations? It would be much appreciated.
Original post by ProStudent
Can someone here explain how to do Simultaneous Equations and Linear Equations? It would be much appreciated.


Is there a specific problem you need help with??
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Dat1Guy
16
A simultaneous equation is often (but could be more) when you have 2 equations where you have 2 variables in each, both containing the same variables, e.g. x and y.

To solve them, you need to eliminate one of the variables, else you can't solve it. To do this, you must make one of the variables the same, for example:
2x + 4y = 10
1x + 5y = 15

You need to choose which variable you want to eliminate. Looking at these 2 equations, it would be easiest to make them both so they have 2x, the 1st one already has 2x, so you need to double the 2nd one so you have:
2x + 10y = 30

so now you have

2x + 4y = 10 and
2x + 10y = 30

You now need some way of removing the x terms, and to do this, you can either make them both equal to 2x:
2x= 10 - 4y
2x= 30 - 10y
and then make them equal to each other:
10 - 4y = 30 - 10y 6y = 20 so y= 20/6 you can now put this back into either of the first equations to solve.

Or, you can take one of the equations from the other:
2x + 10y = 30
- 2x + 4y = 10
6y = 20 so y = 20/6 then you also put this back into either of the original equations
The are two methods - elimination and substitution
Elimination is where you make one of the variable coefficients the same then through addition or subtraction eliminate it
E.g.
1) 2a+3b=21
2) 4a+5b=37
Multiply the first equation by 2 so both the a variables have the same coefficient
1) 4a+6b=42
2) 4a+5b=37
Now subtract the second equation from the first to get
b=5
Substituting b=5 back into one of the equations we can find the value of a as well
2a+3(5)=21
2a+15=21
2a=6
a=3
The second method is by substituting. For one equation, make one of the variables the subject of the equation and substitute it into the other equation
E.g.
1) 2a+3b=21
2)4a+5b=37
Make a the subject of the first equation
1) a=(21-3b)/2
2)4a+5b=37
Now substitute the value of a into the second equation
4((21-3b)/2)+5b=37
Then simplify it
42-6b+5b=37
42-b=37
b=5
And you can again use the value of b to figure if the value of a just like in the first method
When solving two simultaneous equations, one linear and one non-linear, you have to use substitution, and you must substitute the linear equation into the non-linear equation, not the other way round
Hope that helped :smile:
Original post by ProStudent
Can someone here explain how to do Simultaneous Equations and Linear Equations? It would be much appreciated.


let's start with linear equations. ...these are equations with only one unknown for example 2x + 1 = 5...then basically x will be;
2x = 4
x = 2...You just make the x the subject (left x on one side of the equation)

now a simultaneous equation have two equations with the same unknowns..like 2x + y = 10 and 5x + y = 12....You see these are two equation and they can have may difference solutions but we have values that work for both the equations...and to do them we have 3 methods;
elimination
substitution
graph

let's start by elimination..the main aim here is to remove one unknown by either adding or subtracting both the equations like in the example I gave, we can subtract both equations to eliminate y nd then we will only have x in a linear equation and now you know how to solve linear equations...now that you have your x, you just use this x value in one of those two equations and again you will have a linear equation in terms of y and you solve.

substitution:
You just make either y or x the subject in one equation and use this in the second equation like in the example...the first equation can be written as y = 10 - 2x. then use this value of y in the second equation 5x +(10-2x) = 12.now you simplify and you will have a linear equation and you solve the use this value in one of the equations again to solve for the other unknown.

graoh;
this is lengthy but simple....just draw the graphs of both equations and then where they intersect...the coordinates (x coordinate) will be the value of x and y coordinate will be the value of y.


let me know if you understood this :smile:
E.g. 2x + y = 7 and 3x - 2y = 8

1) Try to make either y or x the same.
Here you would make y because one says y and and says 2y so you can simple times y to make two y.

But you have to tines the whole equation by 2.
2x + y = 7 times 2 = 4x + 2y =14

2) put them ontop of eachother
Your left with:
4x + 2y = 14
3x - 2y = 8

(Put the higher = number at the top, 14 is higher than 8)

3) add or subtract
You either add or subtract to get rid of the 2y

In this case you would add because 2y add -2y is 0 and that's what you want

4x + 2y = 14
3x - 2y = 8 +
______________
7x = 22

4) So now you find x
22 divided by 7 = is just 22 over 7
(Leave it as the fraction because the answer would be a decimal and fraction is just easier)
So, X = 22 over 7

5) you have your x so now you find
your y
Get one of the equations and substitute x

E.g. 2x + y = 7
Change x to 22 over 7
22 over 7 times by 2 + y = 7
(These questions will be with a calculator paper)

6) so now you find y
7 - 44 over 7 =5 over 7
Y = 5 over 7


Answers
X=22/7
Y=5/7

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