Q:

Find the magnitude of the acute angle between the lines $2y+3x=4$ and $x+y=5$.

Find the magnitude of the acute angle between the lines $2y+3x=4$ and $x+y=5$.

I have no idea how to start the above equation.
I try to draw the graph of $2y+3x=4$ and $x+y=5$ in the calculator but nothing show in the calculator.

The formula provided in the text book is 
  gradient of $l_{1}=m_{1}=\tan \theta_{1}$
  gradient of $l_{2}=m_{2}=\tan \theta_{2}$

Help me out! thanks.

A:

The equations can be rewritten as:
$$y = -\frac{3}{2}x + 2$$
$$y = -x + 5$$
So, the slopes are: $\tan(\theta_1) =m_1 = -1.5$ and $\tan(\theta_2) = m_2 = -1$.
Therefore, the acute angle($\phi$) between the two lines is:
$$\tan(\phi) = \tan(|\theta_2 - \theta_1|) = \left|\frac{\tan(\theta_2) - \tan(\theta_1)}{1 + \tan(\theta_1)\tan(\theta_2)}\right|$$
or, 
$$\tan(\phi) = \left|\frac{m_2 - m_1}{1 + m_1m_2}\right|$$
$$ = \left|\frac{(-1.5) - (-1)}{1 + (-1.5)(-1)}\right|$$
$$ = \left|-\frac{0.5}{2.5}\right| = \frac{1}{5}$$
Hence, the angle will be:
$$\phi = \tan^{-1}\left(\frac{1}{5}\right) = 11.309932^\circ$$
or $\phi = 11.31^\circ$

