The step-by-step method


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Solving simultaneous equations algebraically

The best way to learn how to do this is to work through an example and practice yourself.

So off we go…

Solve:

2x = 6 – 4y

-3 – 3y = 4x

STEP 1: Label the equations

2x = 6 – 4y        1

-3 – 3y = 4x       2

STEP 2: Rearrange into ax + by = c

In equation 1: -4y becomes+4y

          2x + 4y = 6            1

In equation 2 :  +4x becomes-4x

  •       4x – 3 – 3y = 0       2

and -3 becomes+3

                – 4x – 3y = 3              2

STEP 3: Multiply either or both equations to make either coefficient match

2x becomes 4x    

           4x + 8y = 12         1 x 2

4x remains  4x

 – 4x – 3y = 3         2

STEP 4: Add/subtract the 2 equations to get rid of the equivalent coefficients – in this case, we have a +4x and a -4x, so we need to add them to get to zero (+4 + -4 = 0)

1 + 2:

4x + 8y = 12        +          – 4x – 3y = 3

To get:    

 5y = 15

So:  

 y = 3

But this is only half of the solution – we need the x coordinate too – remember we are finding the point where the two equations cross each other.

STEP 5: Substitute in either of the original equations

Substitute y = 3 into the original equation 1 : 2x = 6 – 4y

To get:      

2x = 6 – 4(3)

which is      

2x =6 – 12

       So:                    

x = -3

STEP 6: OUR SOLUTION

x = -3, y = 3