Note: The answer printed in the book is given as ${\displaystyle ({\frac {7}{8}},{\frac {11}{8}})}$. This is incorrect!

If the two lines ${\displaystyle y_{1}=-2x+5,\,}$ and ${\displaystyle y_{2}=5x-3\,}$ intersect, then ${\displaystyle y_{1}=y_{2}\,}$. Therefore:

${\displaystyle y_{1}-y_{2}=(-2x+5)-(5x-3)=0\,}$

${\displaystyle -7x+8=0\,}$

${\displaystyle -7x=-8\,}$

${\displaystyle x={\frac {8}{7}}}$

Thus, we can now plug in the value for x into any one of our two equations to find the point of interception:

${\displaystyle y_{2}=5\cdot {\frac {8}{7}}-3={\frac {40}{7}}-3={\frac {19}{7}}}$

Thus, the point of interception is ${\displaystyle \left({\frac {8}{7}},{\frac {19}{7}}\right)}$.