Addition method (mathematics)

Illustration of the addition method: From and follows the system of equations . If both scales were in balance beforehand, then the scales are also when you put the respective sides together. ${\displaystyle {\color {Blue}4x}+1={\color {Orange}3y}}$ ${\displaystyle {\color {Orange}2y}={\color {Blue}2x}+2}$ ${\displaystyle {\color {Blue}4x}+1+{\color {Orange}2y}={\color {Orange}3y}+{\color {Blue}2x}+2}$

The addition method is a method that can be used to solve systems of equations . Probably the most well-known approach to solving systems of equations, the Gaussian elimination method , uses the addition method, but it is also of general importance when solving systems of equations.

In the addition method, equations are added . This is usually done by eliminating one or more variables (unknowns) in the equations.

Justification (illustrative)

The following system of linear equations is to be solved as an example :

${\displaystyle {\begin{matrix}(1):\quad &5x&+&3y&=&5\\(2):\quad &2x&-&3y&=&2\end{matrix}}}$

Both equations can be thought of as balanced scales. Scale 1 has to lie in the left bowl and in the right one . Scale 2 has to lie in the left pan and in the right one . ${\displaystyle 5x+3y}$ ${\displaystyle 5}$ ${\displaystyle 2x-3y}$ ${\displaystyle 2}$

If you put the contents of the left bowls together, they must weigh as much as the right bowls combined. The formula obtained is:

${\displaystyle {\begin{matrix}(1)+(2):\quad &(5x+3y)&+&(2x-3y)&=&5+2\end{matrix}}}$

If you sort the left side of the equation according to the unknown, it cancels out and you get a solution for ${\displaystyle y}$ ${\displaystyle x}$

${\displaystyle {\begin{matrix}5x+2x&+&3y-3y&=&7\\7x&+&0&=&7&|:7\\x&&&=&1\end{matrix}}}$

Even multiplying an equation beforehand does not change the balance of the respective balance. A multiple addition process such as or a subtraction process such as is therefore only an abbreviated notation for an equivalent conversion with a subsequent addition process. For the second equation is first tripled and then both equations are added (a detailed example is below). For the second equation is first multiplied by on both sides and then both equations are added. ${\displaystyle (1)+3\cdot (2)}$ ${\displaystyle (1)-(2)}$ ${\displaystyle (1)+3\cdot (2)}$ ${\displaystyle (1)-(2)}$ ${\displaystyle (-1)}$

example

The following system of equations should be solved with the help of the addition method:

${\displaystyle {\begin{matrix}(1)&5x&+&3y&=&5\\(2)&3x&+&y&=&-1\end{matrix}}}$

To do this, one of the two equations has to be transformed in such a way that one variable disappears when the two equations are added together. In this example we multiply equation (2) on both sides by . ${\displaystyle -3}$

${\displaystyle (2)\quad 3x+y=-1|\cdot (-3)}$

This gives us an equivalent system of equations in which the term occurs. ${\displaystyle -3y}$

${\displaystyle {\begin{matrix}(1)&5x&+&3y&=&5\\(2')&-9x&-&3y&=&3\end{matrix}}}$

Now both equations of the system are added and thus summarized in one equation:

${\displaystyle {\begin{matrix}(5x-9x)&+&(3y-3y)&=&5+3\\-4x&+&0y&=&8\\-4x&&&=&8\end{matrix}}}$

Then solve for the remaining variable : ${\displaystyle x}$

${\displaystyle {\begin{matrix}-4x&=&8&|:(-4)\\x&=&-2&\end{matrix}}}$

The value of the first variable is now known. We plug this value ( ) into equation (1) to calculate the value of the second variable. ${\displaystyle x=-2}$

${\displaystyle {\begin{matrix}5\cdot (-2)&+&3y&=&5\\-10&+&3y&=&5&|&+&10\\&&3y&=&15&|&:&3\\&&y&=&5\end{matrix}}}$

This gives us the value for the second variable. The solution of the equation system is available as a solution set to, so . ${\displaystyle \mathbb {L} =\{(-2|5)\}}$

See also

• Appointment procedure , equation procedure