SOLVING SIMULTANEOUS EQUATIONS BY SUBSTITUTION

What are Simultaneous Equations?

Simultaneous Equations are simply TWO Equations which You are required to solve together, OR both Equations being solved with the help of each other. [ Its that simple].

Let us look at these Two Simultaneous Equations, 9a + 8b = 1 and -3a + 6b = 4. As you can see, from these Two Equations, We have Two variables [ a & b]. a & bare called variables because they are unknown. In otherwords, we do not know as yet, what their values are. In that case, we need to solve these Simultaneous Equations to determine their values.

I'm going to use the Substitution method to solve these two Equations and to let you see how easy it is.

So, here we go.......

When solving these Simultaneous Equations, you need to set them out like this:

9a + 8b = 1 ------------ Eqn 1

-3a + 6b = 4 ------------ Eqn 2

By using the substitution method, I'm going to choose one of the Equations and make one of the variables be the subject of the equation.

Lets look at this: If I choose Eqn1 and make [a] the subject of that equation, then We would have an equation like this:

9a = 1 - 8b ------- what I did was to use Eqn1 and take (8b) over the (=) sign to the right of the equation, because 8b is (+), the sign will now change to (-). 1 is already on the right of the equation, so you would have 1 - 8b on the right and 9a on the left of the equation. But we are not yet finish here, because we need to only have (a) on the left of the equation. So, this is what we do:

(9a) is the same as (9Xa) and so, if we want to make (a) the only object to remain on the left of the equation, we would have to remove 9. Therefore, to get rid of 9 from 9a on the left, we have to divide 9a by 9, which is 9a/9. Because it is an equation, what we do to the left, we have to do the same to the right and so, we have to divide 1 - 8b on the right by 9 also, which would now be, (1 - 8b)/9.

Hence, a = (1-8b)/9.

By now, you must be saying, what the HECK is he doing all of this for? I'll show you why.

By going through all of these processes, we can now determine what the numerical value of (b) is, by substituting what we got for (a) into Eqn2 above.

This is how we go about doing it.

Get Eqn2 which is -3a + 6b = 4, look at theEqn and anywhere you see (a) you gonna replace it with (1 - 8b)/9

So, here we go: -3[1-8b]/9 + 6b = 4 If we look carefully at the equation, we will realise that we can now work out the value of the variable b.

Now, the solution is: Use -3 to multiply all that you see in the square brackets [ ]. This would give you a new expression which is -3 + 24b/9, next, add 6b to the expression and equate everything on the left of the equation to 4, which is on the rightof the equation.

This would now be -3 + 24b/9 + 6b = 4.

Now, if you remember what I said earlier above, that because we have an equation, whatever we do to one side, we have to do to the other so as to have both sides balanced.

Now, -3 + 24b is called the numerator and 9 is called the denominator. What we are going to do now, is to simplify this equation in order to determine what the variable b is.

So to get rid of 9 in the denominator, we have to multiply the numerator by the same number which is 9. We also have to multiply 6b by 9 and multiply 4 on the right by 9.

This is what we are left with,

-3 + 24b + 54b = 36,

simplified further,

i -3 + 78b = 36

and simplified further would be

78b = 36+3

which is now,

78b = 39

therefore,

b = 39/78, which is equal to 1/2

Hence the value of the variable b is 1/2

Now that we know what the value of b is, we can now determine the value of a is, by substituting b=1/2 into any one of our two equations.

In this case, let us use Eqn1 for the substitution.

So, 9a + 8(1/2) = 1

9a + 4 = 1

9a = 1 - 4

9a = -3

9a/9 = -3/9

a = -1/3

Therefore, the values of variables a and b are:

a = -1/3 and b= 1/2.

Let me have your comments and all the best.