Functions: Lines and linear functions
Solving systems of equations by addition
A general method of solving two linear equations with two unknowns is by eliminating one unknown. Here we discuss a method that is also suitable for larger systems of linear equations, with more unknowns.
The goal is to give the first equation the form #x=a# and the second equation the form #y=b#.
- The strategy is to edit the equations in such a way that the new system will be equivalent to the old;
- the new system looks more like a solution than the old one.
Steps that will occur are multiplying all terms in the same equation by the same non-zero number and subtracting one equation from the other.
The addition method for linear equations
A system of two linear equations with unknowns #x# and #y# can be solved as follows.
- Make sure #x# occurs in the first equation: if this is not the case, then switch the two equations; this way, #x# will occur in the first equation.
- Replace the second equation by the difference of this equation with a suitable multiple of the first equation, in such a way that #x# no longer occurs in the second equation.
- Replace the first equation by the difference of this equation with a suitable multiple of the second equation in such a way that #y# no longer occurs in the first equation.
- The first equation is now a linear equation with #x# as the only unknown, and the second is a linear equation with #y# as the only unknown. These equations can be solved with the theory of linear equations with one unknown.
For the system it is assumed that #x# and #y# really occur in the system.
- If only #x# occurs, then we are dealing with a system of equations with one unknown which has been dealt with earlier. For each solution #x=a# of that system, and every real number #b#, we have the solution #xa\land y=b# to the system (and these are all solutions).
- If only #y# occurs, then the same observations hold, with #x# and #y# interchanged.
- If both #x# and #y# do not occur, then each pair #\rv{x,y}# is a solution if all equations are satisfied (think of #0=0#), and not a single pair is a solution if at least one of the equations is a contradiction (like #0=1#).
Solving by addition
This method is known as the addition method.
After all, you mostly add a multiple of one equation to another.
It may be that after the second step, the second equation becomes true (if #0=0#) or a contraction (if #0=1#) because not only #x# but also #y# disappears. In that case, the solution is a line given by the first equation.
There are many ways to get to this solution. We will describe one.
- To make sure that the unknown #x# is present in the first equation, we switch the two equations if this was not the case in the original system: # -4\cdot x+6\cdot y-4=0\land x-y-2=0 #.
- Next we get rid of the term with #x# from the second equation by multiplying the first equation with #\frac{1}{-4}# and subtracting from the second: # -4\cdot x+6\cdot y-4=0\land {{y}\over{2}}=0 #.
- By dividing (left and right hand side) in the second equation by #{{1}\over{2}}# we find #y=6#. We now have the system # -4\cdot x+6\cdot y-4=0\land y=6 #.
- If we enter the solution of #y# (the second equation) in the first equation (or stated different: we subtract #6# times the second equation from the first), then we find the system: # 32-4\cdot x=0\land y=6 #.
- The first equation can be solved as discussed in in Solving by reduction of a linear equation with one unknown. The result is #x= 8\land y = 6#.
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