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Simultaneous equations

Method 1- elimination

- The object is to eliminate either the x's or the y's.

- Do this by adding or subtracting the equations from each other

- Try to work through these examples

2x + y = 3
x - y = 1
2x + y = 3
x + y = 1
Add
3x = 4
x= 11/3
Subtract
x = 2
Now find y

11/3 - y = 1

11/3 - 1 = y

1/3 = y

Now find y

2 + y = 1

y = -1

Check your answer by substituting both x and y into the first equation Check your answer by substituting both x and y into the first equation

Sometimes, you may need to rearrange the equations before adding or subtracting

2x + y = 3    (1)
x - 3y = 7     (2)

If you simply add or subtract one of these from the other, neither x nor y will be eliminated. Therefore you have to either multiply the first equation by 3 or the second by 2 before you can proceed.

 

Method 2- substitution

1) Rearrange one equation to get either y = _____ or x = _____

2) Substitute it onto the other equation

3) Solve the equation!

4) Use one of the original equations to work out the other letter (x or y) that you do not already have

5) Check your answers in the other  equation

2x + y = 3

x - y = 1

---------------

1) y = 3 - 2x

2) x - (3 - 2x) = 1

3) x - 3 + 2x = 1

    3x - 3 = 1

    3x = 4

    x = 4/3

    x = 11/3

4) 2(11/3) + y = 3

    22/3 + y = 3

    y = 3 -  22/3

    y = 1/3

5) 11/3 - 1/3 = 1

    1 = 1 ... so it worked!

 

 

Non-linear simultaneous equations

Use the method of Substitution

x y = 2        (1)

x - y = 1      (2)

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                                                                                          Answer:    x = y + 1    (2)

                                                         (y + 1) y = 2     [(1) with (2) substituted into it]

                                                        y2 + y = 2        [It is now a normal quadratic]

y2 + y - 2 = 0

(y + 1)(y + 2) = 0

y = 1 or -2

Now find x

x = y + 1

x = 2 or -1

 

 

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