### Video Transcript

Express the simultaneous equations
three π₯ plus two π¦ is equal to 12 and three π₯ plus π¦ is equal to seven as a
matrix equation.

So weβre given a set of two
simultaneous equations in two variables and asked to express this set of equations
in matrix form. And what this requires us to do is
to separate out the coefficients of our variables into one matrix and the variables
themselves into another and the constants on the right-hand side into a third. And because we have two equations
in two variables π₯ and π¦, our coefficient matrix will be a two-by-two matrix, that
is, with two rows and two columns. Our variables matrix will be a
two-by-one column matrix, and our constants again, a two-by-one column matrix.

Itβs important to note that we must
be able to reform our original equations by performing matrix multiplication on the
left-hand side. And remember that multiplying a
matrix with π rows and π columns by a matrix with π rows and π columns will give
us a matrix with π rows and π columns. And for matrix multiplication to
work, the number of columns π in our first matrix must be the number of rows in the
second matrix.

So now we want to find the matrix
equation in this form which, if we multiply it out, reproduces our original system
of equations. So letβs look more closely at our
equations. Itβs important to make sure before
we start putting entries into our matrix that the variables are aligned in our
system of equations so that the π₯βs are above one another and so are the π¦βs. The reason for this is that when we
populate our matrix, weβre going to read off the coefficients of the π₯βs and
π¦βs.

Our coefficients in the first
equation, equation one, are three and two with the constant 12 on the right-hand
side and the three and two from the first row in our coefficient matrix and the 12,
the first element in our matrix on the right-hand side. Similarly, the elements in the
second row of our coefficient matrix are the coefficients in the second equation,
equation two, that is, three and one. And the second element on our
matrix of constant on the right-hand side is seven. This matrix equation is the full
matrix representation of the set of simultaneous equations three π₯ plus two π¦ is
equal to 12 and three π₯ plus π¦ is equal to seven.

If we were to apply matrix
multiplication to the left-hand side of our matrix equation, we would have three π₯
plus two π¦ is equal to 12, which is our first equation one, and three π₯ plus π¦ is
equal to seven, which is our second equation. And weβre back to our original set
of simultaneous linear equations.