Simultaneous Equations

Of the 2 equations given; one will be linear and the other quadratic.

Examples of linear equations
(does not contain “squares”)
a + b = 2
2m + n = -1
2x – 3y = 5

Examples of quadratic equations
(contains “squares)
4x² + 2y² - 2x + 3y = 10
x² + 4xy + 2y² = 5

Now let us look at a typical example:
Solve for a and b simultaneously:
a² + b² - 6a -6b = 7 (eq.1)
a + 3b =17 (eq.2)

SOLUTION
It is generally easier to work with the linear equation first.
In this case, equation 2.
We are solving for the “easier” of the 2 variables. i.e.

a = 17-3b (eq3)

We must now substitute this into the first equation as follows.
This means that wherever you see a, substitute it with (17-3b).

We now re-arrange this long story by grouping the like terms:

We must simplify:
10b² – 90b + 187 = 7
10b² – 90b + 187 – 7= 0
10b² – 90b + 180 = 0

Since the numbers in the trinomial above are big, we can make our task easier by dividing by 10. So we will now have:
b² – 9b + 18 = 0
(b-3) (b-6) = 0

From here we can see that b will have 2 values:
b = 3 or b= 6.

Therefore substitute into the linear equation i.e. equation 3.

Remember that
a = 17 - 3b

Therefore:
When b=3
a = 17-3(3)
a = 17 - 9
a = 8

When b= 6
a = 17-3(6)
a = 17 – 18
a = -1

As a check substitute the following:
a=8 and b=3
into the original equations (Equations 1 and 2) and confirm that these values are indeed correct!

Do the same for a=-1 and b= 6.