Opposite Coefficients Method for Solving Two Equations in Two Unknowns

coefficients of the first equation

4

4

4

3

3

coefficients of the second equation

6

6

3

3

2

2

To eliminate x (step 1),

multiply the first equation by g and

the second equation by h,

then add the two equations (step 2).

Automatic elimination:

g

3

h

-2

show steps

1

2

3

Solve the system of the equations:

4x+4y3

6x+3y2

12x+12y9

-12x-6y-4

6y5

y

5

6

4x+

10

3

3

x-

1

12

This Demonstration solves the system of linear equations

ax+by=c

,

dx+ey=f

with the opposite coefficient method.

Step 1: to eliminate

x

, multiply the first equation by

d

and the second by

-a

, and add the equations. Even better: multiply the first with

g=lcm(a,d)/a

and the second with

h=-lcm(a,d)/d

. (Here

lcm(a,d)

means the least common multiple of

a

and

d

.)

Step 2: add equations and solve a new equation for

y

.

Step 3: substitute the solution for

y

into the first given equation, and solve the new equation for

x

.