Answer Key 3.6

$m=2$
$m=-\dfrac{2}{3}$
$m=4$
$m=-10$
$\begin{array}{rrrrlrrr} \\ \\ \\ \\ x&-&y&=&4&&& \\ -x&&&&-x&&& \\ \midrule &&(-y&=&-x&+&4)&(-1) \\ &&y&=&x&-&4& \\ &&m&=&1&&& \end{array}$
$\begin{array}{rrrrlrrr} \\ \\ \\ \\ \\ \\ \\ \\ 6x&-&5y&=&20&&& \\ -6x&&&&-6x&&& \\ \midrule &&\dfrac{-5y}{-5}&=&\dfrac{-6x}{-5}&+&\dfrac{20}{-5}& \\ \\ &&y&=&\dfrac{6}{5}x&-&4& \\ \\ &&m&=&\dfrac{6}{5}&&& \end{array}$
$\begin{array}{rrlrrr} \\ \\ \\ \\ \\ y&=&\dfrac{1}{3}x&&& \\ \\ \therefore m&=&\dfrac{1}{3} &&& \\ m_{\perp} &=&-1&\div &\dfrac{1}{3}&\text{or} \\ m_{\perp}&=&-3 &&& \end{array}$
$\begin{array}{lrlrrrr} \\ \\ \\ \\ m&=&-\dfrac{1}{2} &&&& \\ m_{\perp} &=&-1&\div &-\dfrac{1}{2}&&\\ \\ m_{\perp}&=&-1 &\cdot &-\dfrac{2}{1}&=& 2 \end{array}$
$\begin{array}{lrlrrrr} \\ \\ \\ \\ m&=&-\dfrac{1}{3} &&&& \\ m_{\perp} &=&-1&\div &-\dfrac{1}{3}&&\\ \\ m_{\perp}&=&-1 &\cdot &-\dfrac{3}{1}&=& 3 \end{array}$
$\begin{array}{lrlrrrr} \\ \\ \\ \\ m&=&\dfrac{4}{5} &&&& \\ m_{\perp} &=&-1&\div &\dfrac{4}{5}&&\\ \\ m_{\perp}&=&-1 &\cdot &\dfrac{5}{4}&=& -\dfrac{5}{4} \end{array}$
$\begin{array}{rrrrlrr} \\ \\ \\ \\ \\ \\ \\ \\ x&-&3y&=&-6& \\ -x&&&&-x&& \\ \midrule &&\dfrac{-3y}{-3}&=&\dfrac{-x}{-3}&-&\dfrac{6}{-3} \\ \\ &&y&=&\dfrac{1}{3}x&+&2 \\ &&m_{\perp}&=&-1&\div &\dfrac{1}{3} \\ &&m_{\perp}&=&-3&& \end{array}$
$\begin{array}{rrrrrrr} \\ \\ \\ \\ \\ \\ 3x&-&y&=&-3& \\ -3x&&&&-3x&& \\ \midrule &&-y&=&-3x&-&3 \\ &&y&=&3x&+&3 \\ &&m_{\perp}&=&-1&\div &3 \\ &&m_{\perp}&=&-\dfrac{1}{3}&& \end{array}$
$\begin{array}{rrrrlrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ m&=&\dfrac{2}{5}&&&& \\ \\ y&-&y_1&=&m(x&-&x_1) \\ y&-&4&=&\dfrac{2}{5}(x&-&1) \\ \\ y&-&4&=&\dfrac{2}{5}x&-&\dfrac{2}{5} \\ \\ &+&4&&&+&4 \\ \midrule &&y&=&\dfrac{2}{5}x&+&\dfrac{18}{5} \end{array}$
$\begin{array}{rrrrlrr} \\ \\ \\ \\ \\ \\ m&=&-3&&&& \\ \\ y&-&y_1&=&m(x&-&x_1) \\ y&-&2&=&-3(x&-&5) \\ y&-&2&=&-3x&+&15 \\ &+&2&&&+&2 \\ \midrule &&y&=&-3x&+&17 \end{array}$
$\begin{array}{rrrrlrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ m&=&\dfrac{1}{2}&&&& \\ \\ y&-&y_1&=&m(x&-&x_1) \\ y&-&4&=&\dfrac{1}{2}(x&-&3) \\ \\ y&-&4&=&\dfrac{1}{2}x&-&\dfrac{3}{2} \\ \\ &+&4&&&+&4 \\ \midrule &&y&=&\dfrac{1}{2}x&+&\dfrac{5}{2} \end{array}$
$\begin{array}{rrrrlrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ m&=&\dfrac{4}{3}&&&& \\ \\ y&-&y_1&=&m(x&-&x_1) \\ y&-&-1&=&\dfrac{4}{3}(x&-&1) \\ \\ y&+&1&=&\dfrac{4}{3}x&-&\dfrac{4}{3} \\ \\ &-&1&&&-&1 \\ \midrule &&y&=&\dfrac{4}{3}x&-&\dfrac{7}{3} \end{array}$
$\begin{array}{rrrrlrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ m&=&-\dfrac{3}{5}&&&& \\ \\ y&-&y_1&=&m(x&-&x_1) \\ y&-&3&=&-\dfrac{3}{5}(x&-&2) \\ \\ y&-&3&=&-\dfrac{3}{5}x&+&\dfrac{6}{5} \\ \\ &+&3&&&+&3 \\ \midrule &&y&=&-\dfrac{3}{5}x&+&\dfrac{21}{5} \end{array}$
$\begin{array}{rrrrlrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ m&=&\dfrac{1}{3}&&&& \\ \\ y&-&y_1&=&m(x&-&x_1) \\ y&-&3&=&\dfrac{1}{3}(x&-&-1) \\ \\ y&-&3&=&\dfrac{1}{3}x&+&\dfrac{1}{3} \\ \\ &+&3&&&+&3 \\ \midrule &&y&=&\dfrac{1}{3}x&+&\dfrac{10}{3} \end{array}$
$\begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -x&+&y&=&1&&&& \\ +x&&&&+x&&&& \\ \midrule &&y&=&x&+&1&& \\ &&\therefore m&=&1&&&& \\ \\ y&-&y_1&=&m(x&-&x_1)&& \\ y&-&-5&=&1(x&-&1)&& \\ y&+&5&=&x&-&1&& \\ -y&-&5&&-y&-&5&& \\ \midrule &&0&=&x&-&y&-&6 \end{array}$
$\begin{array}{rrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -x&+&2y&=&2&&& \\ +x&&&&+x&&& \\ \midrule &&2y&=&x&+&2& \\ &\text{or}&y&=&\dfrac{1}{2}x&+&1& \\ \\ &&\therefore m&=&-2&&& \\ \\ y&-&y_1&=&m(x&-&x_1)& \\ y&-&-2&=&-2(x&-&1)& \\ y&+&2&=&-2x&+&2& \\ -y&-&2&&-y&-&2& \\ \midrule &&(0&=&-2x&-&y)&(-1) \\ &&0&=&2x&+&y& \end{array}$
$\begin{array}{rrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 5x&+&y&=&-3&&&&& \\ -5x&&&&-5x&&&&& \\ \midrule &&y&=&-5x&-&3&&& \\ &&\therefore m&=&-5&&&&& \\ \\ y&-&y_1&=&m(x&-&x_1)&&& \\ y&-&2&=&-5(x&-&5)&&& \\ y&-&2&=&-5x&+&25&&& \\ -y&+&2&&-y&+&2&&& \\ \midrule &&(0&=&-5x&-&y&+&27)&(-1) \\ &&0&=&5x&+&y&-&27& \end{array}$
$\begin{array}{rrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -x&+&y&=&1&&&&& \\ +x&&&&+x&&&&& \\ \midrule &&y&=&x&+&1&&& \\ &&\therefore m&=&-1&&&&& \\ \\ y&-&y_1&=&m(x&-&x_1)&&& \\ y&-&3&=&-1(x&-&1)&&& \\ y&-&3&=&-x&+&1&&& \\ -y&+&3&&-y&+&3&&& \\ \midrule &&(0&=&-x&-&y&+&4)&(-1) \\ &&0&=&x&+&y&-&4& \end{array}$
$\begin{array}{rrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -4x&+&y&=&0&&&& \\ +4x&&&&+4x&&&& \\ \midrule &&y&=&4x&&&& \\ &&\therefore m&=&4&&&& \\ \\ y&-&y_1&=&m(x&-&x_1)&& \\ y&-&2&=&4(x&-&4)&& \\ y&-&2&=&4x&-&16&& \\ -y&+&2&&-y&+&2&& \\ \midrule &&0&=&4x&-&y&-&14 \end{array}$
$\begin{array}{rrrrrrrrrr} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3x&+&7y&=&0&&&&& \\ -3x&&&&-3x&&&&& \\ \midrule &&7y&=&-3x&&&&& \\ &\text{or}&y&=&-\dfrac{3}{7}x&&&&& \\ \\ &&\therefore m&=&\dfrac{7}{3}&&&&& \\ \\ y&-&y_1&=&m(x&-&x_1)&&& \\ y&-&-5&=&\dfrac{7}{3}(x&-&-3)&&& \\ \\ y&+&5&=&\dfrac{7}{3}x&+&7&&& \\ \\ -y&-&5&&-y&-&5&&& \\ \midrule &&(0&=&\dfrac{7}{3}x&-&y&+&2)&(3) \\ \\ &&0&=&7x&-&3y&+&6& \end{array}$
$y=-3$
$x=-5$
$x=-3$
$y=0$
$y=-1$
$x=2$
$x=-2$
$y=-4$
$y=3$
$x=-3$
$x=5$
$y=-1$

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