### 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.

\[ {\lineqs{ -8 x+9 y +3 &=& 0 \cr x -y +5 &=& 0 \cr}} \]

Give the answer in the form #x=a\land y=b# for suitable values of #a# and #b#.

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: # -8\cdot x+9\cdot y+3=0\land x-y+5=0 #.
- Next we get rid of the term with #x# from the second equation by multiplying the first equation with #\frac{1}{-8}# and subtracting from the second: # -8\cdot x+9\cdot y+3=0\land {{y}\over{8}}=0 #.
- By dividing (left and right hand side) in the second equation by #{{1}\over{8}}# we find #y=-43#. We now have the system # -8\cdot x+9\cdot y+3=0\land y=-43 #.
- If we enter the solution of #y# (the second equation) in the first equation (or stated different: we subtract #9# times the second equation from the first), then we find the system: # -8\cdot x-384=0\land y=-43 #.
- The first equation can be solved as discussed in in
*Solving by reduction of a linear equation with one unknown*. The result is #x= -48\land y = -43#.

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