Linear Simultaneous Equations¶

To solve linear simultaneous equation, you will need to solve the equation for one variable - and then calculate for the other.

Given two different linear equations, you could use either of the two example approaches below to solve.

Solving by elimination¶

Solving by elimination involves attempting to remove a variable by adding/subtracting the equations. This is accomplished by matching the coefficient of the same variable in both equations.

Example¶

$7x + y = 9 \\ 2x + 3y = 8$

Match 2 coefficients¶

For the sake of simplicity, match the $y$ coefficient

$\begin{align} 7x + y & = 9 \\ 21x + \underbrace{3y}_\text{see? it matches} & = 27 \\ \\ 2x + \underbrace{3y}_\text{it already does} & = 27 \end{align}$

The $y$ coefficients match.

Add / Subtract Equations¶

Take a look at the operation before $y$ , and use the following table to establish whether to add or subtract the equations.

Operation	Add	Subtract
Add	Subtract	Add
Subtract	Add	Subtract

Using the above table, you can establish that the example needs to be subtracted.

$\begin{align} 21x + 3y & = 27 \\ - 2x + 3y & = 8 \\ 19x & = 19 \\ x & = 1 \end{align}$

Solving by substitution¶

Solving by substitution involves substituting one equation into the other to eliminate a variable. This typically involves rearranging an equation.

Example¶

$7x + y = 9 \\ 2x + 3y = 8$

Rearrange ready for substitution¶

Since it's the easiest to isolate, I will substitute to find $y$ from the first equation.

$7x + y = 9 \\ \Rightarrow y = 9 - 7x$

Substitute into other equation (and find x)¶

$\begin{align} & 2x + 3y = 8 \\ \Rightarrow & 2x + 3(9 - 7x) = 8 \\ \Rightarrow & 2x + 27 - 21x = 8 \\ \Rightarrow & 27 - 8 = 21x - 2x \\ \Rightarrow & 19 = 19x \\ \Rightarrow & x = 1 \end{align}$

Solve for second variable¶

To establish $y$ you will need to substitute the $x$ value back into one of the original equations.

$\begin{align} 7(1) + y & = 9 \\ 7 + y & = 9 \\ \\ y & = 2 \end{align}$

Except as otherwise noted, the content of this page is licensed under the Creative Commons Attribution 4.0 License, and code samples are licensed under the Mozilla Public License 2.0.