Question Video: Expressing a Pair of Simultaneous Equations as a Matrix Equation Mathematics • First Year of Secondary School

Video Transcript

Express the simultaneous equations three 𝑥 minus 24 is equal to negative eight 𝑦 and 𝑥 is equal to three minus 𝑦 as a matrix equation.

In order to answer this question, we need to ensure that both of our equations are written in standard form. We need to rewrite the first equation so it is in the form 𝑎𝑥 plus 𝑏𝑦 is equal to 𝑝 and the second equation so it is in the form 𝑐𝑥 plus 𝑑𝑦 is equal to 𝑞, where 𝑎, 𝑏, 𝑐, 𝑑 as well as 𝑝 and 𝑞 are constants.

Let’s begin with the equation three 𝑥 minus 24 is equal to negative eight 𝑦. We can add 24 and eight 𝑦 to both sides of our equation. This gives us three 𝑥 plus eight 𝑦 is equal to 24. This equation has now been written in standard form. Our second equation stated that 𝑥 is equal to three minus 𝑦. Adding 𝑦 to both sides of this equation and ensuring that our 𝑥- and 𝑦-terms are in the same order gives us 𝑥 plus 𝑦 is equal to three.

We now have a pair of linear simultaneous equations written in standard form. Our matrix equation will consist of a two-by-two coefficient matrix. The coefficients of 𝑥 and 𝑦 in our first equation are three and eight. These will make up the top row. The coefficients of 𝑥 and 𝑦 in our second equation are one and one. We therefore have the two-by-two coefficient matrix three, eight, one, one.

Our variables are 𝑥 and 𝑦. This means that the coefficient matrix can be multiplied by this column variable matrix. On the right-hand side of our two equations, we have the constants 24 and three. Our simultaneous equations can be expressed as the matrix equation three, eight, one, one multiplied by 𝑥, 𝑦 is equal to 24, three.