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.