Solve the following systems of

Question

Solve the following systems of equations by matrix method $$2ax-2by=-a-4b$$ and $$2bx+2ay=4a-b$$

I only need the equation in terms of $x$ and $y$ in order to represent in the matrix form but how can I eliminate $a$ and $b$.

You already have the equations in terms of $x$ and $y$. – amd Jun 18 '16 at 05:38 — amd, Jun 18 '16 at 05:38

score 1 · Answer 1 · answered Jun 18 '16 at 05:33

1

I don't see the point why you need it in the form of only $x$ and $y$ since you can write and row reduce the following:

$$ \begin{bmatrix}2a & -2b & -a-4b \\2b & 2a & 4a-b \end{bmatrix}$$

answered Jun 18 '16 at 05:33

Joshua

I need coefficients in one matrix and variables in another and the product of these two matrix must be equal to another matrix i.e in the form of AX=B and X=A^(-1)B. – user335710 Jun 18 '16 at 05:36
Clearly, from your problem, $A$ matrix here is $ \begin{bmatrix}2a & -2b \2b & 2a \end{bmatrix}$. But in order for the necessary condition for $X = A^{-1}B$ to exist, you'll need to have an invertible matrix. – Joshua Jun 18 '16 at 05:41
The simplest invertible matrix you can think of, would be the identity matrix. In that case, you'll have to row reduce the above matrix, and you'll get the identity matrix on the left side of the augmented matrix, with the right side of the matrix being the one with variables $a$ and $b$ – Joshua Jun 18 '16 at 05:43

score 1 · Answer 2 · answered Jun 18 '16 at 05:55

Take $2a$ times (both sides of) your first equation and add it to $2b$ times the second equation:

$$4(a^2+b^2)x=-2a^2-2b^2$$

Divide to solve for $x$.

Take $2b$ times your first equation and subtract $2a$ times the second equation:

$$-4(a^2+b^2)y=-8a^2-8b^2$$

Divide to solve for $y$.

2 Answers2