Solve (simple linear) equations with matrices

Let us have two simple, linear equations and two unknowns; something like:
Equation 1: 7x + 3y = 44

Equation 2: -1x + 10y = 25

There are a few ways to solve for both x and y.

We can, of course multiply one or both equations such that when we subtract one result for the other, one of the unknowns goes away. For example:

Eq. 1: 7x + 3y = 44

Eq. 2: -1x + 10y = 25 —> multiply Eq. 2 by -7 —>

Eq. 2': 7x - 70y = -175

—> Subtract Eq. 1 from Eq. 2':

(7x - 7x) + (-70y - 3y) = (-175 - 44) —>

-73y = -219 —>

y = 219 / 73 = 3 —>

And since 7x + 3y = 44 —>

7x = 44 - 9 = 35 —>

x = 35 / 7 = 5

Another way to solve (most instances) equations such as these is by using matrices like so:

We set up two matrices: one matrix — let us call that one left— containing the values on the left side of the "=" sign of the equations. 7 3 -1 10 and one matrix — let us call that one right— holding the values to the right of the "=" sign: 44 25 Next, we create a third matrix which is the so-called inverse of the left matrix1: 0.137 -0.041 0.014 0.096 Finally, to solve for x and y we multiply the inverse matrix with the right matrix2. This gives a matrix with the solutions for x (top) and y (bottom): 0.137 -0.041 0.014 0.096 × 44 25 = 5 3 Note: not all cases of two simple linear equations can be solved this way. For instance, when the left sides of the equations are identical or when they are a multiple of each other, the inverse matrix cannot be computed.

To play with this matrix solution follow these steps: Create two equations of the form Px + Qy = Result Fill in the values for P, Q and Result in the text boxes below. Hit the Solve for x and y! button to have the browser do the matrix math. Equation 1: x + y = Equation 2: x + y = Solve for x and y! left matrix: 7 3 -1 10 right matrix: 44 25 inverse of the left matrix: 0.137 -0.041 0.014 0.096 inverse × right matrix = solutions matrix with x (top) and y (bottom): 0.137 -0.041 0.014 0.096 × 44 25 = 5 3

¹Here we do not go into how to create this inverse matrix.
²Note that matrix multiplication is (generally) not commutative; i.e., A × B ≠ B × A.