# Question Video: Identifying a Pair of Simultaneous Equations from a Matrix Equation Mathematics • 10th Grade

Write down the set of simultaneous equations that could be solved using the given matrix equation. [3, 3 and 2, 4][𝑎 and 𝑏] = [10 and 12]

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### Video Transcript

Write down the set of simultaneous equations that could be solved using the given matrix equation, that is, where the two-by-two matrix with elements three, three, two, four multiplied by the column matrix with elements 𝑎, 𝑏 is equal to the column matrix with elements 10, 12.

We’re given a matrix equation of the form 𝐴𝑋 is equal to 𝐵. That’s where 𝐴 is the matrix of coefficients, the coefficient matrix; 𝑋 is the variable matrix, that’s the matrix of unknowns; and 𝐵, the matrix on the right-hand side, is the constant matrix.

Now, in order to recover the set of simultaneous equations that could be solved using the matrix equation given, we need to multiply the coefficient matrix 𝐴 by the variable matrix 𝑋. We then equate each line of our result with the appropriate line of the matrix 𝐵. So now let’s multiply out our left-hand side. Multiplying our first row with the column matrix 𝑎, 𝑏, we have three times 𝑎 plus three 𝑏. And equating to the first row in our column matrix 𝐵, that’s equal to 10.

Next, we multiply our second row by our column matrix 𝑋. And we have two multiplied by 𝑎 plus four multiplied by 𝑏. And this equates to the element in the second row of matrix 𝐵, which is 12.

The set of simultaneous equations that could be solved using the given matrix equation is therefore three 𝑎 plus three 𝑏 is equal to 10 and two 𝑎 plus four 𝑏 is equal to 12.