Simultaneous Equations

These are pairs of equations that relate to each other such as:

$\begin{aligned} 2x + y &= 5 &\color{blue}{\text{ (1) }} \\ 4x + 3y - 4 & = 20 &\color{blue}{\text{ (2) }} \end{aligned}$

It's also probably a good idea to label each equation so you don't get confused.

For example the $x$ value and $y$ value are the same in both equations. In order to find them we can use a few different strategies.

Substitution

This method works by rearranging one equation and then adding it to the other to remove an unknown term. For example:

$\begin{aligned} 2x \color{red}{(-2x)} + y &= 5 \color{red}{(-2x)} &\color{blue}{\text{ (1) }} \\ y &= 5 - 2x \end{aligned}$

Now that we have found the value of $y$ in terms of the other unknown numbers we can substitute it into $\color{blue}{\text{ (2) }}$, the other equation:

$\begin{aligned} 4x + 3(\color{red}{5 - 2x}) - 4 &= 20 &\color{blue}{\text{ (1) into (2) }} \\ \end{aligned}$

Now we can solve this equation like a regular algebra question:

$\begin{aligned} 4x + 15 - 6x - 4&= 20&\color{blue}{\text{ (2) }} \\ -2x + 11\color{red}{(-20 + 2x)} &= 20 \color{red}{(-20 + 2x)} \\[7mu] \dfrac{- 9}{\color{red}{2}} &= \dfrac{2x}{\color{red}{2}} \\[7mu] -4.5 &= x \end{aligned}$

Now that we have a number equal to $x$ we can substitute the value into 1 of the equations:

$\begin{aligned} 2(\color{red}{-4.5}) + y&= 5&\color{blue}{\text{ (2) into (1) }} \\ -9 \color{red}{+9} + y &= 5 \color{red}{+9} \\ y &= 14 \end{aligned}$

Like all of algebra, you should check your answers by putting them back into the original equations:

$\begin{aligned} 2(\color{red}{-4.5}) + \color{red}{14}&= 5&\color{blue}{\text{ (1) check }} \\ 4(\color{red}{-4.5}) + 3(\color{red}{14}) - 4&= 5&\color{blue}{\text{ (2) check }} \\ \end{aligned}$

Elimination

Graphing