Adding the second equation to the first corresponds to adding the same quantity to both sides of
the first equation. Thus, the resulting equation is still true. Similarly, two equations can be
Subtracting the second equation from the first corresponds to subtracting the same quantity from
both sides of the first equation. Thus, the resulting equation is still true.
The basic approach used to solve a system of equations is to reduce the system by eliminating
the unknowns one at a time until one equation with one unknown results. This equation is solved
and its value used to determine the values of the other unknowns, again one at a time. There are
three different techniques used to eliminate unknowns in systems of equations: addition or
subtraction, substitution, and comparison.
Solving Simultaneous Equations
The simplest system of equations is one involving two linear equations with two unknowns.
5x + 6y = 12
3x + 5y = 3
The approach used to solve systems of two linear equations involving two unknowns is to
combine the two equations in such a way that one of the unknowns is eliminated. The resulting
equation can be solved for one unknown, and either of the original equations can then be used
to solve for the other unknown.
Systems of two equations involving two unknowns can be solved by addition or subtraction using
Multiply or divide one or both equations by some factor or factors that
will make the coefficients of one unknown numerically equal in both
Eliminate the unknown having equal coefficients by addition or
Solve the resulting equation for the value of the one remaining unknown.