Systems of linear equations

Systems of linear equations: Two equations with two unknowns

Systems of linear equations

Assume that for two given unknown numbers #x# and #y#, it is known that they are solutions to the following linear equations:

\[\lineqs{3 x +2 y +1&=&0\cr -2 x -3 y -6&=&0\cr}\]

We call this a system of two equations with two unknowns. The solution to the system is the point #\rv{x,y}# which is a solution to both linear equations. Hence, the solution to the system is the intersection point of the two lines which represent the linear equations.

We can also write the system with the sign for "and", which is #\land#. It then looks like this:

\[\begin{array}{rcl} 3 x +2 y +1=0 &\land&-2 x -3 y -6=0\end{array}\]

geogebra plaatje

Solution

The solution to

\[\lineqs{3 x +2 y +1&=&0\cr -2 x -3 y -6&=&0\cr}\]

is #\blue x=\blue{\tfrac{9}{5}}# and #\green y=\green{-\tfrac{16}{5}}#, because #3\cdot \blue{\tfrac{9}{5}} +2\cdot \green{-\tfrac{16}{5}} +1=0# and #-2\cdot \blue{\tfrac{9}{5}} -3 \cdot \green{-\tfrac{16}{5}}-6=0#

Later we will see how we can determine the solution to a system in a systematic way.

Take a look at the following system of linear equations \[\lineqs{9\cdot x+4\cdot y+4=0 \cr -2\cdot x+7\cdot y+3=0 \cr}\] Is the point #\rv{-2, {{7}\over{2}}}# a solution of the system?

If #\rv{-2, {{7}\over{2}}}# is a solution, then both equations have to be true if we substitute #x=-2# and #y={{7}\over{2}}# in the equations.
In this case, if we substitute #x=-2# and #y={{7}\over{2}}# in the system, we get:
\[\lineqs{9\cdot -2+4\cdot {{7}\over{2}}+4=0 \cr -2\cdot -2+7\cdot {{7}\over{2}}+3={{63}\over{2}} \ne 0 \cr}\]
Only the first equation is correct, but the second one is not, hence #\rv{-2, {{7}\over{2}}}# is not a solution of the system.

New example