Solving simultaneous equations can be done quite easily by elimination.
However, what happens if substitution is investigated ?
Solving the equations
ax +by = e
cx + dy = f
Note, the numerator is found by cross multiplying
and subtracting using the y co-efficients.
bf de
The denominator is found by reverse cross multiplying
and subtracting the coefficients of x and y.
bc ad
And, for y
Note, the numerator is found by cross multiplying
and subtracting using the x co-efficients.
af ce
The denominator is found by cross multiplying
and subtracting the coefficients of x and y.
ad bc
This is the vedic way of solving simultaneous equations.
It will only work when ad – bc ≠ 0
Example, solve the following system of equations
2x +3y = 7
3x + 4y = 6
Example, solve the following system of equations
2x +3y = 7
3x + 4y = 6
Click for a spreadsheet to calculate a 2x2 system.
The system of equations
ax + by = e
cx + dy = f
can be written in matrix form as
Ax = b
Where
Notice that the denominator from above,
ad – bc , is found by cross multiplying and subtracting
the elements in A.
This is known as the determinant of A.
It is written det(A) or
If det(A) = 0, the system has no solution.
Examples
Does the follow system of equations
have a solution ?
3x + 2y = 7
9x + 6y = 6
No solution, since determinant = 0
Each element in A is associated with its minor
To find the determinant,
Place signs
Example
Find the determinant of
Note:
Cofactors are minors with their place sign.
Example
Find the cofactors of
which has
First row
Second row
Third row
Combining these with the will give a new matrix C,
made up of the cofactors of A.
so
This is written Adj (A)
It is the transpose of C
so
Adj(A) = CT
Example
Find Adj (A) for the matrix
From the work above,
Transposed,
Example
Find the inverse of the matrix
From the work above,
Note,
Example, using the matrix from above
This means that Gaussian elimination can be used to find the inverse:
Write down the matrix to be inverted with the Identity matrix next to it.
Perform ERO’s to reduce the left hand matrix to I,
simultaneously performing the same ERO’s on the Identity matrix.
The result is the inverse of the given matrix.
Example – using the question given above.