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

Express the simultaneous equations 3𝑎 + 2𝑏 = 13, 2𝑎 + 3𝑏 = 7 as a matrix equation.

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Video Transcript

Express the simultaneous equations three 𝑎 plus two 𝑏 equals 13 and two 𝑎 plus three 𝑏 equal seven as a matrix equation.

We recall that we can represent a system of linear equations in matrix form using a coefficient matrix, a variable matrix, and a constant matrix. Before doing this, we need to ensure that our equations are written in standard form, 𝑎𝑥 plus 𝑏𝑦 is equal to 𝑐, where 𝑎, 𝑏, and 𝑐 are constants. In this question, both of our equations are written in standard form. However, it is important to note that 𝑎 and 𝑏 are variables and not constants.

We know that the coefficient matrix can be formed by aligning the coefficients of the variables of each equation in a row. The coefficients of our first equation are three and two. In the second equation, we have two and three, giving us the two-by-two coefficient matrix three, two, two, three. The variables here are 𝑎 and 𝑏. So we can write the variable matrix as the two-by-one matrix 𝑎, 𝑏. On the right-hand side, we have the constant terms 13 and seven. As these correspond to the first and second equation, respectively, the constant matrix is 13, seven.

We now have a matrix equation made up of a coefficient matrix, a variable matrix, and a constant matrix. The simultaneous equations three 𝑎 plus two 𝑏 is equal to 13 and two 𝑎 plus three 𝑏 is equal to seven can be rewritten as a matrix equation three, two, two, three multiplied by 𝑎, 𝑏 is equal to 13, seven.

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