Respuesta :
if the line containing the coordinate "k", is parallel to the line at 5,5 and 1,-4, then their slopes must be the same, since parallel lines, have the same slope, let's check both slopes then.
[tex]\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 5}}\quad ,&{{ 5}})\quad % (c,d) &({{ 1}}\quad ,&{{ -4}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{-4-5}{1-5}\implies \cfrac{-9}{-4}\implies \boxed{\cfrac{9}{4}}\\\\ -------------------------------\\\\[/tex]
[tex]\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ -5}}\quad ,&{{ k}})\quad % (c,d) &({{ 2}}\quad ,&{{ 10}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{10-k}{2-(-5)}\implies \cfrac{10-k}{2+5} \\\\\\ \stackrel{\textit{parallel lines, same slope}}{\cfrac{10-k}{7}=\boxed{\cfrac{9}{4}}}\implies 40-4k=63\implies -23=4k\implies -\cfrac{23}{4}=k[/tex]
[tex]\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 5}}\quad ,&{{ 5}})\quad % (c,d) &({{ 1}}\quad ,&{{ -4}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{-4-5}{1-5}\implies \cfrac{-9}{-4}\implies \boxed{\cfrac{9}{4}}\\\\ -------------------------------\\\\[/tex]
[tex]\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ -5}}\quad ,&{{ k}})\quad % (c,d) &({{ 2}}\quad ,&{{ 10}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{10-k}{2-(-5)}\implies \cfrac{10-k}{2+5} \\\\\\ \stackrel{\textit{parallel lines, same slope}}{\cfrac{10-k}{7}=\boxed{\cfrac{9}{4}}}\implies 40-4k=63\implies -23=4k\implies -\cfrac{23}{4}=k[/tex]
