Download Simultaneous Linear Equations – A General Solution

Simultaneous Linear Equations – A General Solution
Mathematical Mischief – January 2013
Question - Take the following simultaneous linear equations:
ax + by = c
dx + ey = f
Devise a general solution for x and y .
ax + by = c
ax + by =
ax
=
x
=
dx + ey = f
dx + ey =
c
c − by
dx
c − by
x
a
c − by f − ey
=
a
d
f
=
f − ey
=
f − ey
d
c − by
=
a
d(c − by) =
f − ey
d
a( f − ey)
cd − bdy
=
af − aey
cd − af
=
bdy − aey
cd − af
= (bd − ae)y
cd − af
bd − ae
=
Step 1: State both equations
separately.
Step 2: Solve each equation for x .
Step 3: Substitute x for the
solution of the other equation.
Step 4: Solve for y .
y
As it would be inappropriate to substitute the value of y into one
equation to solve for x , it is far more reasonable to start again, solving
the simultaneous equations from the beginning.
ax + by =
by
=
y
=
c
c − ax
dx + ey =
ey
c − ax
y
b
c − ax f − dx
=
b
e
f
=
f − dx
=
f − dx
e
Step 5: Solve each equation for y .
Step 6: Substitute y for the
solution of the other equation.
January 2013 – http://www.mathematicalmischief.com/ - @mathmischief
c − ax
=
b
e(c − ax) =
f − dx
e
b( f − dx)
ce − aex
=
bf − bdx
ce − bf
=
aex − bdx
ce − bf
= (ae − bd)x
ce − bf
ae − bd
=
Step 7: Solve for x .
x
As a result, when solving two simultaneous equations of the form:
ax + by = c
dx + ey = f
We obtain a general solution that:
x=
ce − bf
ae − bd
and
y=
cd − af
bd − ae
January 2013 – http://www.mathematicalmischief.com/ - @mathmischief