Simultaneous Equations

When you have an equation with one unknown, such as 3x+7=19, you can solve it.

But if you have an equation with two unknowns, such as y=3x+7, then you can't.

      You need another, simultaneous equation, such as y = 9x+13.

       Now you can subtract one from the other and as y-y = 0 you get

                                                                               0 = 6x+6 which you can solve

                                                                              -6 = 6x

                                                                              -1 = x

             And you put this into one of the original equations to work out y

                                                                              y = 3(-1)+7

                                                                              y = -3+7

                                                                              y = 4

Your equations may come in the form   y-3x = 7    and    y-9x = 13

but the method is exactly the same.

                                        y-9x = 13

                                        y-3x = 7 Remember that when you subtract a negative number you add it which gives you

                                          -6x = 6

                                                    and then the same as before.

5F  Solve these pairs of simultaneous equations.

1.   3x+2y=7                2.  6x+y=22                3.  5x+3y=18

     5x+2y=9                      3x+y=13                      x+3y=12

In the questions above the coefficient of y (the number in front of the y) is the same.  This may not always be the case,

3x+2y = 16

5x - 6y = 8

9x+6y = 48

14x = 56

   x = 4

3(4) + 2y = 16

          2y = 16 - 12

             y=2

 Here both the coefficients of x and the coefficients of y are different.  So it is necessary to multiply one, or both equations so that either the x or the y coefficients are the same.

Multiplying the first equation by 3 makes it

And because we now have a +6y in one equation and a - 6y in the other, the equations need to be ADDED to each other, not subtracted

 

And now y can be worked out, using the first equation as in the simpler examples.

 

 

4.  5x+3y = 11                5.  4x+4y = 36                6.  10x - 8y = 7

       2x+y = 4                      3x -2y = 2                         6x+4y = 25

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