Answer Key 3.5

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

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Intermediate Algebra (Convert to MathJax) Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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