Question Video: Expressing a Pair of Simultaneous Equations as a Matrix Equation Mathematics

Video Transcript

Express the simultaneous equations three 𝑥 plus two 𝑦 is equal to 12 and three 𝑥 plus 𝑦 is equal to seven as a matrix equation.

So we’re 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 𝑥 and 𝑦, 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.

It’s 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 𝑚 rows and 𝑛 columns by a matrix with 𝑛 rows and 𝑝 columns will give us a matrix with 𝑚 rows and 𝑝 columns. And for matrix multiplication to work, the number of columns 𝑛 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 let’s look more closely at our equations. It’s 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 𝑥’s are above one another and so are the 𝑦’s. The reason for this is that when we populate our matrix, we’re going to read off the coefficients of the 𝑥’s and 𝑦’s.

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 𝑥 plus two 𝑦 is equal to 12 and three 𝑥 plus 𝑦 is equal to seven.

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