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

Express the simultaneous equations (1/3)𝑥 − (2/3)𝑦 = 5/3, (3/4)𝑦 + (1/4)𝑥 = 7/4 as a matrix equation.


Video Transcript

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.

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