Question

Solve the system of linear equations for x and y.(cos θ)x + (sin θ)y=1(−sin θ)x + (cos θ)y = 0x=y=

Answer 1

Answer:

$\bold{x =cos\theta}\\\bold{y=sin\theta}$

Step-by-step explanation:

The given system of linear equations is:

$(cos\theta)x+(sin\theta)y=1\\(-sin\theta)x+(cos\theta)y=0$

We have to solve the equations for the values of $x, y$.

Let us use elimination method in which we eliminate one of the variables from the two variables.

For this, let us multiply the first equation by $sin\theta$ and second equation by $cos\theta$

Now, the equations become:

$(cos\theta.sin\theta)x+(sin\theta.sin\theta)y=sin\theta\\\Rightarrow (cos\theta.sin\theta)x+(sin^2\theta)y=sin\theta ....... (1)\\\\(-sin\theta.cos\theta)x+(cos\theta.cos\theta)y=0\\\Rightarrow (-sin\theta.cos\theta)x+(cos^2\theta)y=0 ..... (2)$

Now, let us add (1) and (2):

$(sin^2\theta)y+(cos^2\theta)y=sin\theta\\\Rightarrow (sin^2\theta+cos^2\theta)y=sin\theta\\\Rightarrow (1)y=sin\theta\\\Rightarrow y = sin\theta$

Using the equation:

$(-sin\theta)x+(cos\theta)y=0$

Putting value of $y$:

$\Rightarrow (-sin\theta)x+(cos\theta)sin\theta=0\\\Rightarrow (sin\theta)x=(cos\theta)sin\theta\\\Rightarrow x = cos\theta$

So, the answer to the system of linear equations is:

$\bold{x =cos\theta}\\\bold{y=sin\theta}$