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Answer Key 3.5

  1. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&3&=&\dfrac{2}{3}(x&-&2) \\ \\
    y&-&3&=&\dfrac{2}{3}x&-&\dfrac{4}{3} \\ \\
    &+&3&&&+&3 \\
    \hline
    &&y&=&\dfrac{2}{3}x&+&\dfrac{5}{3}
    \end{array}\)
  2. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&2&=&4(x&-&1) \\
    y&-&2&=&4x&-&4 \\
    &+&2&&&+&2 \\
    \hline
    &&y&=&4x&-&2
    \end{array}\)
  3. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&2&=&\dfrac{1}{2}(x&-&2) \\ \\
    y&-&2&=&\dfrac{1}{2}x&-&1 \\
    &+&2&&&+&2 \\
    \hline
    &&y&=&\dfrac{1}{2}x&+&1
    \end{array}\)
  4. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&1&=&-\dfrac{1}{2}(x&-&2) \\ \\
    y&-&1&=&-\dfrac{1}{2}x&+&1 \\
    &+&1&&&+&1 \\
    \hline
    &&y&=&-\dfrac{1}{2}x&+&2
    \end{array}\)
  5. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-5&=&9(x&-&-1) \\
    y&+&5&=&9x&+&9 \\
    &-&5&&&-&5 \\
    \hline
    &&y&=&9x&+&4
    \end{array}\)
  6. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-2&=&-2(x&-&2) \\
    y&+&2&=&-2x&+&4 \\
    &-&2&&&-&2 \\
    \hline
    &&y&=&-2x&+&2
    \end{array}\)
  7. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&1&=&\dfrac{3}{4}(x&-&-4) \\ \\
    y&-&1&=&\dfrac{3}{4}x&+&3 \\
    &+&1&&&+&1 \\
    \hline
    &&y&=&\dfrac{3}{4}x&+&4
    \end{array}\)
  8. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-3&=&-2(x&-&4) \\
    y&+&3&=&-2x&+&8 \\
    &-&3&&&-&3 \\
    \hline
    &&y&=&-2x&+&5
    \end{array}\)
  9. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-2&=&-3(x&-&0) \\
    y&+&2&=&-3x&& \\
    &-&2&&&-&2 \\
    \hline
    &&y&=&-3x&-&2 \\
    \end{array}\)
  10. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&1&=&4(x&-&-1) \\
    y&-&1&=&4x&+&4 \\
    &+&1&&&+&1 \\
    \hline
    &&y&=&4x&+&5
    \end{array}\)
  11. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-5&=&-\dfrac{1}{4}(x&-&0) \\ \\
    y&+&5&=&-\dfrac{1}{4}x&& \\
    &-&5&&&-&5 \\
    \hline
    &&y&=&-\dfrac{1}{4}x&-&5
    \end{array}\)
  12. \(\begin{array}{rrrrlrr}
    \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&2&=&-\dfrac{5}{4}(x&-&0) \\ \\
    y&-&2&=&-\dfrac{5}{4}x&& \\
    &+&2&&&+&2 \\
    \hline
    &&y&=&-\dfrac{5}{4}x&+&2
    \end{array}\)
  13. \(\begin{array}{rrrrlrrrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1)&& \\
    y&-&-5&=&2(x&-&-1)&& \\
    y&+&5&=&2x&+&2&& \\
    -y&-&5&&-y&-&5&& \\
    \hline
    &&0&=&2x&-&y&-&3
    \end{array}\)
  14. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1)&&& \\
    y&-&-2&=&-2(x&-&2)&&& \\
    y&+&2&=&-2x&+&4&&& \\
    -y&-&2&&-y&-&2&&& \\
    \hline
    &&(0&=&-2x&-&y&+&2)&(-1) \\
    &&0&=&2x&+&y&-&2&
    \end{array}\)
  15. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1)&&& \\
    y&-&-1&=&-\dfrac{3}{5}(x&-&5)&&& \\ \\
    y&+&1&=&-\dfrac{3}{5}x&+&3&&& \\ \\
    -y&-&1&&-y&-&1&&& \\
    \hline
    &&(0&=&-\dfrac{3}{5}x&-&y&+&2)&(-5) \\ \\
    &&0&=&3x&+&5y&-&10&
    \end{array}\)
  16. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-2&=&-\dfrac{2}{3}(x&-&-2) \\ \\
    y&+&2&=&-\dfrac{2}{3}x&-&\dfrac{4}{3} \\ \\
    -y&-&2&&-y&-&2 \\
    \hline
    &&(0&=&-\dfrac{2}{3}x&-&y&-&\dfrac{10}{3})&(-3) \\ \\
    &&0&=&2x&+&3y&+&10&
    \end{array}\)
  17. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&1&=&\dfrac{1}{2}(x&-&-4) \\ \\
    y&-&1&=&\dfrac{1}{2}x&+&2 \\ \\
    -y&+&1&&-y&+&1 \\
    \hline
    &&(0&=&\dfrac{1}{2}x&-&y&+&3)&(2) \\ \\
    &&0&=&x&-&2y&+&6&
    \end{array}\)
  18. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-3&=&-\dfrac{7}{4}(x&-&4) \\ \\
    y&+&3&=&-\dfrac{7}{4}x&+&7 \\ \\
    -y&-&3&&-y&-&3 \\
    \hline
    &&(0&=&-\dfrac{7}{4}x&-&y&+&4)&(-4) \\ \\
    &&0&=&7x&+&4y&-&16&
    \end{array}\)
  19. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-2&=&-\dfrac{3}{2}(x&-&4) \\ \\
    y&+&2&=&-\dfrac{3}{2}x&+&6 \\ \\
    -y&-&2&&-y&-&2 \\
    \hline
    &&(0&=&-\dfrac{3}{2}x&-&y&+&4)&(-2) \\ \\
    &&0&=&3x&+&2y&-&8&
    \end{array}\)
  20. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&0&=&-\dfrac{5}{2}(x&-&-2) \\ \\
    &&y&=&-\dfrac{5}{2}x&-&5 \\ \\
    &&-y&&-y&& \\
    \hline
    &&(0&=&-\dfrac{5}{2}x&-&y&+&5)&(-2) \\ \\
    &&0&=&5x&+&2y&+&10&
    \end{array}\)
  21. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-3&=&-\dfrac{2}{5}(x&-&-5) \\ \\
    y&+&3&=&-\dfrac{2}{5}x&-&2 \\ \\
    -y&-&3&&-y&-&3 \\
    \hline
    &&(0&=&-\dfrac{2}{5}x&-&y&-&5)&(-5) \\ \\
    &&0&=&2x&+&5y&+&25&
    \end{array}\)
  22. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&3&=&\dfrac{7}{3}(x&-&3) \\ \\
    y&-&3&=&\dfrac{7}{3}x&-&7 \\ \\
    -y&+&3&&-y&+&3 \\
    \hline
    &&(0&=&\dfrac{7}{3}x&-&y&-&4)&(3) \\ \\
    &&0&=&7x&-&3y&-&12&
    \end{array}\)
  23. \(\begin{array}{rrrrlrrrr}
    \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1)&& \\
    y&-&-2&=&1(x&-&2)&& \\
    y&+&2&=&x&-&2&& \\
    -y&-&2&&-y&-&2&& \\
    \hline
    &&0&=&x&-&y&-&4
    \end{array}\)
  24. \(\begin{array}{rrrrlrrrrr}
    \\ \\ \\ \\ \\ \\ \\ \\ \\ \\
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&4&=&-\dfrac{1}{3}(x&-&-3) \\ \\
    y&-&4&=&-\dfrac{1}{3}x&-&1 \\ \\
    -y&+&4&&-y&+&4 \\
    \hline
    &&(0&=&-\dfrac{1}{3}x&-&y&+&3)&(-3) \\ \\
    &&0&=&x&+&3y&-&9&
    \end{array}\)
  25. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{1-3}{-3--4}\Rightarrow \dfrac{-2}{1}\Rightarrow -2 \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&1&=&-2(x&-&-3) \\
    y&-&1&=&-2x&-&6 \\
    &+&1&&&+&1 \\
    \hline
    &&y&=&-2x&-&5
    \end{array}\)
  26. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{-3-3}{-3-1}\Rightarrow \dfrac{-6}{-4}\Rightarrow \dfrac{3}{2} \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&3&=&\dfrac{3}{2}(x&-&1) \\ \\
    y&-&3&=&\dfrac{3}{2}x&-&\dfrac{3}{2} \\ \\
    &+&3&&&+&3 \\
    \hline
    &&y&=&\dfrac{3}{2}x&+&\dfrac{3}{2}
    \end{array}\)
  27. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{0-1}{-3-5}\Rightarrow \dfrac{-1}{-8}\Rightarrow \dfrac{1}{8} \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&0&=&\dfrac{1}{8}(x&-&-3) \\ \\
    &&y&=&\dfrac{1}{8}x&+&\dfrac{3}{8}
    \end{array}\)
  28. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{4-5}{4--4}\Rightarrow \dfrac{-1}{8} \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&4&=&-\dfrac{1}{8}(x&-&4) \\ \\
    y&-&4&=&-\dfrac{1}{8}x&+&\dfrac{1}{2} \\ \\
    &+&4&&&+&4 \\
    \hline
    &&y&=&-\dfrac{1}{8}x&+&\dfrac{9}{2}
    \end{array}\)
  29. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{4--2}{0--4}\Rightarrow \dfrac{6}{4}\Rightarrow \dfrac{3}{2} \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&4&=&\dfrac{3}{2}(x&-&0) \\ \\
    y&-&4&=&\dfrac{3}{2}x&& \\
    &+&4&&&+&4 \\
    \hline
    &&y&=&\dfrac{3}{2}x&+&4
    \end{array}\)
  30. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{4-1}{4--4}\Rightarrow \dfrac{3}{8} \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&4&=&\dfrac{3}{8}(x&-&4) \\ \\
    y&-&4&=&\dfrac{3}{8}x&-&\dfrac{3}{2} \\ \\
    &+&4&&&+&4 \\
    \hline
    &&y&=&\dfrac{3}{8}x&+&\dfrac{5}{2}
    \end{array}\)
  31. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{3-5}{-5-3}\Rightarrow \dfrac{-2}{-8}\Rightarrow \dfrac{1}{4} \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&3&=&\dfrac{1}{4}(x&-&-5) \\ \\
    y&-&3&=&\dfrac{1}{4}x&-&\dfrac{5}{4} \\ \\
    &+&3&&&+&3 \\
    \hline
    &&y&=&\dfrac{1}{4}x&+&\dfrac{17}{4}
    \end{array}\)
  32. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{0--4}{-5--1}\Rightarrow \dfrac{4}{-4}\Rightarrow -1 \\ \\ \)
    \(\begin{array}{rrrrlrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&0&=&-1(x&-&-5) \\
    &&y&=&-x&-&5
    \end{array}\)
  33. \[
    m=\frac{\Delta y}{\Delta x}
    =\frac{5-(-3)}{-4-3}
    =\frac{8}{-7}
    =-\frac{8}{7}
    \]

    \[
    \begin{array}{rrrrlrrrrr}
    y &- & y_1 &=& m(x &- & x_1) \\[6pt]
    y &- & 5 &=& -\frac{8}{7}(x &- & -4) \\[6pt]
    y &- & 5 &=& -\frac{8}{7}x &- & \frac{32}{7} \\[6pt]
    -y &+& 5 && -y &+& 5 \\ \hline
    && 0 &=& \frac{8}{7}x + y - \frac{3}{7} \\[6pt]
    && 0 &=& 8x + 7y - 3
    \end{array}
    \]

  34. \[
    m=\frac{\Delta y}{\Delta x}
    =\frac{-4-(-5)}{-5-(-1)}
    =\frac{1}{-4}
    =-\frac{1}{4}
    \]

    \[
    \begin{array}{rrrrlrrrrr}
    y &- & y_1 &=& m(x &- & x_1) \\[6pt]
    y &- & (-5) &=& -\frac14(x - (-1)) \\[6pt]
    y &+ & 5 &=& -\frac14(x + 1) \\[6pt]
    y &+ & 5 &=& -\frac14 x &- & \frac14 \\[6pt]
    -y &- & 5 && -y &- & 5 \\ \hline
    && 0 &=& -\frac14 x - y - \frac{21}{4} \\[6pt]
    && 0 &=& x + 4y + 21
    \end{array}
    \]

  35. \[
    m=\frac{\Delta y}{\Delta x}
    =\frac{4-(-3)}{-2-3}
    =\frac{7}{-5}
    =-\frac75
    \]

    \[
    \begin{array}{rrrrlrrrrr}
    y &- & y_1 &=& m(x &- & x_1) \\[6pt]
    y &- & 4 &=& -\frac75(x - (-2)) \\[6pt]
    y &- & 4 &=& -\frac75 x &- & \frac{14}{5} \\[6pt]
    -y &+ & 4 && -y &+ & 4 \\ \hline
    && 0 &=& -\frac75 x - y + \frac65 \\[6pt]
    && 0 &=& 7x + 5y - 6
    \end{array}
    \]

  36. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{-4--7}{-3--6}\Rightarrow \dfrac{3}{3}\Rightarrow 1 \\ \\ \)
    \(\begin{array}{rrrrlrrrr}
    y&-&y_1&=&m(x&-&x_1)&& \\
    y&-&-4&=&1(x&-&-3)&& \\
    y&+&4&=&x&+&3&& \\
    -y&-&4&&-y&-&4&& \\
    \hline
    &&0&=&x&-&y&-&1
    \end{array}\)
  37. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{-2-1}{-1--5}\Rightarrow \dfrac{-3}{4} \\ \\ \)
    \(\begin{array}{rrrrlrrrrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-2&=&-\dfrac{3}{4}(x&-&-1) \\ \\
    y&+&2&=&-\dfrac{3}{4}x&-&\dfrac{3}{4} \\ \\
    -y&-&2&&-y&-&2 \\
    \hline
    &&(0&=&-\dfrac{3}{4}x&-&y&-&\dfrac{11}{4}) &(-4) \\ \\
    &&0&=&3x&+&4y&+&11&
    \end{array}\)
  38. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{-2--1}{5--5}\Rightarrow \dfrac{-1}{10} \\ \\ \)
    \(\begin{array}{rrrrlrrrrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-1&=&-\dfrac{1}{10}(x&-&-5) \\ \\
    y&+&1&=&-\dfrac{1}{10}x&-&\dfrac{1}{2} \\ \\
    -y&-&1&&-y&-&1 \\
    \hline
    &&(0&=&-\dfrac{1}{10}x&-&y&-&\dfrac{3}{2}) &(-10) \\ \\
    &&0&=&x&+&10y&+&15&
    \end{array}\)
  39. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{-3-5}{2--5}\Rightarrow \dfrac{-8}{7} \\ \\ \)
    \(\begin{array}{rrrrlrrrrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-3&=&-\dfrac{8}{7}(x&-&2) \\ \\
    y&+&3&=&-\dfrac{8}{7}x&+&\dfrac{16}{7} \\ \\
    -y&-&3&&-y&-&3 \\
    \hline
    &&(0&=&-\dfrac{8}{7}x&-&y&-&\dfrac{5}{7}) &(-7) \\ \\
    &&0&=&8x&+&7y&+&5&
    \end{array}\)
  40. \(\phantom{1}\)
    \(m=\dfrac{\Delta y}{\Delta x}\Rightarrow \dfrac{-4--1}{-5-1}\Rightarrow \dfrac{-3}{-6}\Rightarrow \dfrac{1}{2} \\ \\ \)
    \(\begin{array}{rrrrlrrrrr}
    y&-&y_1&=&m(x&-&x_1) \\
    y&-&-1&=&\dfrac{1}{2}(x&-&1) \\ \\
    y&+&1&=&\dfrac{1}{2}x&-&\dfrac{1}{2} \\ \\
    -y&-&1&&-y&-&1 \\
    \hline
    &&(0&=&\dfrac{1}{2}x&-&y&-&\dfrac{3}{2}) &(2) \\ \\
    &&0&=&x&-&2y&-&3&
    \end{array}\)

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