Solving systems of equations by addition

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.

Solve the following system of equations with unknowns $x$ and $y$ .

${\lineqs{ 7 x+9 y -4 &=& 0 \cr -3 x -4 y +1 &=& 0 \cr}}$

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

$x= 7\land y = -5$

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: $7\cdot x+9\cdot y-4=0\land -3\cdot x-4\cdot y+1=0$ .
Next we get rid of the term with $x$ from the second equation by multiplying the first equation with $\frac{-3}{7}$ and subtracting from the second: $7\cdot x+9\cdot y-4=0\land -{{y}\over{7}}=0$ .
By dividing (left and right hand side) in the second equation by $-{{1}\over{7}}$ we find $y=-5$ . We now have the system $7\cdot x+9\cdot y-4=0\land y=-5$ .
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: $7\cdot x-49=0\land y=-5$ .
The first equation can be solved as discussed in in Solving by reduction of a linear equation with one unknown. The result is $x= 7\land y = -5$ .

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