Express the simultaneous equations
one-third 𝑥 minus two-thirds 𝑦 is equal to five-thirds and three-quarters 𝑦 plus
one-quarter 𝑥 is equal to seven-quarters as a matrix equation. In order to rewrite a system of
linear equations as a matrix equation, we need to find a coefficient matrix, a
variable matrix, and a constant matrix. Before starting, however, we need
to ensure that all of our equations are written in standard form.
As the 𝑥-term comes first in our
first equation, we can rewrite the second equation as a quarter 𝑥 plus
three-quarters 𝑦 is equal to seven-quarters. This is because addition is
commutative. The coefficients of our first
equation are one-third and negative two-thirds, whilst the coefficients of our
second equation are one-quarter and three-quarters. This means that the two-by-two
coefficient matrix is one-third, negative two-thirds, one-quarter, three
quarters. The two variables here are 𝑥 and
𝑦. Therefore, the variable matrix is
simply 𝑥, 𝑦.
On the right-hand side of our
equations, we have the constant terms five-thirds and seven-quarters. These make up the constant
matrix. We now have a matrix equation made
up of a coefficient matrix, a variable matrix, and a constant matrix as
required. The two simultaneous equations
expressed as a matrix equation is one-third, negative two-thirds, one-quarter,
three-quarters multiplied by 𝑥, 𝑦 is equal to five-thirds, seven-quarters.