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

Express the given simultaneous equations as a matrix equation. 4𝑥 − 2𝑦 = 0, 3𝑦 + 5𝑥 = −11.

02:28

Video Transcript

Express the given simultaneous equations as a matrix equation. So we have four 𝑥 minus two 𝑦 is equal to zero. And three 𝑦 plus five 𝑥 is equal to negative 11.

So when we’re gonna represent it as a matrix equation, we’re gonna represent it in this form. So I’ve just highlighted what’s gonna go in the first matrix. So in our first matrix, what we’ll have is, the first column is our 𝑥-coefficients. And our second column is going to be our 𝑦-coefficients. And we know that the first matrix is gonna be a two-by-two matrix cause we’ve got two columns because we have two unknowns, 𝑥 and 𝑦. And we have two rows because we actually have two equations. So we’ve got four 𝑥 minus two 𝑦 equals zero and three 𝑦 plus five 𝑥 equals negative 11.

Okay, great. So that’s the first matrix in our matrix equation. The second matrix actually is our unknowns. So this is where our 𝑥 and 𝑦 will go. And our final matrix is our solutions. Okay, well great. We’ve now got the form for our matrix equation. Let’s set up our matrix equation for our simultaneous equations.

Okay, so our first column is going to be four, five. And the reason it’s that is cause they are the coefficients of our 𝑥 terms. As you can see, I’ve actually underlined them. And I’ve actually underlined the sign in front of the bottom one, which is positive five. The reason I’ve done that is to make sure that, you know, that you do take into account the sign of the coefficient.

Also, it’s worth noting here that they’re actually not aligned. So we can see here that the second term is the 𝑥 term in the second equation. That doesn’t matter. You always have the coefficients of 𝑥 wherever they are. And our second column is going to be a negative two, three. And again, I’ve paid attention to the signs. So it’s negative two. And then this is all multiplied by the matrix 𝑥𝑦. And that’s because these are our variables. And then finally, this is all equal to the matrix zero, negative 11. And that’s because these are the solutions to our equations.

Great. So I’ve actually reached our final solution. And we can say that the simultaneous equations four 𝑥 minus two 𝑦 equals zero and three 𝑦 plus five 𝑥 equals negative 11 can be shown as a matrix equation, where the matrix four, negative two, five, three multiplied by the matrix 𝑥𝑦 is equal to the matrix zero, negative 11.

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