### Video Transcript

Express the simultaneous equations
three 𝑎 plus two 𝑏 equals 13 and two 𝑎 plus three 𝑏 equal seven as a matrix
equation.

We recall that we can represent a
system of linear equations in matrix form using a coefficient matrix, a variable
matrix, and a constant matrix. Before doing this, we need to
ensure that our equations are written in standard form, 𝑎𝑥 plus 𝑏𝑦 is equal to
𝑐, where 𝑎, 𝑏, and 𝑐 are constants. In this question, both of our
equations are written in standard form. However, it is important to note
that 𝑎 and 𝑏 are variables and not constants.

We know that the coefficient matrix
can be formed by aligning the coefficients of the variables of each equation in a
row. The coefficients of our first
equation are three and two. In the second equation, we have two
and three, giving us the two-by-two coefficient matrix three, two, two, three. The variables here are 𝑎 and
𝑏. So we can write the variable matrix
as the two-by-one matrix 𝑎, 𝑏. On the right-hand side, we have the
constant terms 13 and seven. As these correspond to the first
and second equation, respectively, the constant matrix is 13, seven.

We now have a matrix equation made
up of a coefficient matrix, a variable matrix, and a constant matrix. The simultaneous equations three 𝑎
plus two 𝑏 is equal to 13 and two 𝑎 plus three 𝑏 is equal to seven can be
rewritten as a matrix equation three, two, two, three multiplied by 𝑎, 𝑏 is equal
to 13, seven.