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Solve the following linear equations for x and y: 3x − 4 = 5 − 2x, 2x − 1 = x + 5, y + 10 = 30, y + 11 = 11, y − 3 = 10, 8 - y = -12

Mathematics

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  1. Identify the equation.

    3x−4=5−2x3x - 4 = 5 - 2x

  2. Add $2x$ to both sides of the equation.

    3x−4+2x=5−2x+2x3x - 4 + 2x = 5 - 2x + 2x 5x−4=55x - 4 = 5

  3. Add 4 to both sides of the equation.

    5x−4+4=5+45x - 4 + 4 = 5 + 4 5x=95x = 9

  4. Divide both sides by 5.

    5x5=95\frac{5x}{5} = \frac{9}{5} x=95x = \frac{9}{5}

    The solution is:

    x=95\boxed{x = \frac{9}{5}}

    :

  5. Identify the equation.

    2x−1=x+52x - 1 = x + 5

  6. Subtract xx from both sides of the equation.

    2x−1−x=x+5−x2x - 1 - x = x + 5 - x x−1=5x - 1 = 5

  7. Add 1 to both sides of the equation.

    x−1+1=5+1x - 1 + 1 = 5 + 1 x=6x = 6

    The solution is:

    x=6\boxed{x = 6}

    :

  8. Identify the equation.

    y+10=30y + 10 = 30

  9. Subtract 10 from both sides of the equation.

    y+10−10=30−10y + 10 - 10 = 30 - 10 y=20y = 20

    The solution is:

    y=20\boxed{y = 20}

    :

  10. Identify the equation.

    y+11=11y + 11 = 11

  11. Subtract 11 from both sides of the equation.

    y+11−11=11−11y + 11 - 11 = 11 - 11 y=0y = 0

    The solution is:

    y=0\boxed{y = 0}

    :

  12. Identify the equation.

    y−3=10y - 3 = 10

  13. Add 3 to both sides of the equation.

    y−3+3=10+3y - 3 + 3 = 10 + 3 y=13y = 13

    The solution is:

    y=13\boxed{y = 13}

    :

  14. Identify the equation.

    8−y=−128 - y = -12

  15. Subtract 8 from both sides of the equation.

    8−y−8=−12−88 - y - 8 = -12 - 8 −y=−20-y = -20

  16. Multiply both sides by -1.

    (−1)(−y)=(−1)(−20)(-1)(-y) = (-1)(-20) y=20y = 20

    The solution is:

    y=20\boxed{y = 20}

    :

  17. Identify the equation.

    y+12=10y + 12 = 10

  18. Subtract 12 from both sides of the equation.

    y+12−12=10−12y + 12 - 12 = 10 - 12 y=−2y = -2

    The solution is:

    y=−2\boxed{y = -2}

    :

  19. Identify the equation.

    y+10=12y + 10 = 12

  20. Subtract 10 from both sides of the equation.

    y+10−10=12−10y + 10 - 10 = 12 - 10 y=2y = 2

    The solution is:

    y=2\boxed{y = 2}

    :

  21. Identify the equation.

    y+19=25y + 19 = 25

  22. Subtract 19 from both sides of the equation.

    y+19−19=25−19y + 19 - 19 = 25 - 19 y=6y = 6

    The solution is:

    y=6\boxed{y = 6}

    :

  23. Identify the equation.

    6+2x=15−x6 + 2x = 15 - x

  24. Add xx to both sides of the equation.

    6+2x+x=15−x+x6 + 2x + x = 15 - x + x 6+3x=156 + 3x = 15

  25. Subtract 6 from both sides of the equation.

    6+3x−6=15−66 + 3x - 6 = 15 - 6 3x=93x = 9

  26. Divide both sides by 3.

    3x3=93\frac{3x}{3} = \frac{9}{3} x=3x = 3

    The solution is:

    x=3\boxed{x = 3}

    :

  27. Identify the equation.

    3x+7=15−x3x + 7 = 15 - x

  28. Add xx to both sides of the equation.

    3x+7+x=15−x+x3x + 7 + x = 15 - x + x 4x+7=154x + 7 = 15

  29. Subtract 7 from both sides of the equation.

    4x+7−7=15−74x + 7 - 7 = 15 - 7 4x=84x = 8

  30. Divide both sides by 4.

    4x4=84\frac{4x}{4} = \frac{8}{4} x=2x = 2

    The solution is:

    x=2\boxed{x = 2}

    :

  31. Identify the equation.

    5+x=11−2x5 + x = 11 - 2x

  32. Add $2x$ to both sides of the equation.

    5+x+2x=11−2x+2x5 + x + 2x = 11 - 2x + 2x 5+3x=115 + 3x = 11

  33. Subtract 5 from both sides of the equation.

    5+3x−5=11−55 + 3x - 5 = 11 - 5 3x=63x = 6

  34. Divide both sides by 3.

    3x3=63\frac{3x}{3} = \frac{6}{3} x=2x = 2

    The solution is:

    x=2\boxed{x = 2}

    :

  35. Identify the equation.

    17−3x=4−x17 - 3x = 4 - x

  36. Add $3x$ to both sides of the equation.

    17−3x+3x=4−x+3x17 - 3x + 3x = 4 - x + 3x 17=4+2x17 = 4 + 2x

  37. Subtract 4 from both sides of the equation.

    17−4=4+2x−417 - 4 = 4 + 2x - 4 13=2x13 = 2x

  38. Divide both sides by 2.

    132=2x2\frac{13}{2} = \frac{2x}{2} x=132x = \frac{13}{2}

    The solution is:

    x=132\boxed{x = \frac{13}{2}}

    :

  39. Identify the equation.

    8−x=x+68 - x = x + 6

  40. Add xx to both sides of the equation.

    8−x+x=x+6+x8 - x + x = x + 6 + x 8=2x+68 = 2x + 6

  41. Subtract 6 from both sides of the equation.

    8−6=2x+6−68 - 6 = 2x + 6 - 6 2=2x2 = 2x

  42. Divide both sides by 2.

    22=2x2\frac{2}{2} = \frac{2x}{2} x=1x = 1

    The solution is:

    x=1\boxed{x = 1}

    :

  43. Identify the equation.

    7x+9=237x + 9 = 23

  44. Subtract 9 from both sides of the equation.

    7x+9−9=23−97x + 9 - 9 = 23 - 9 7x=147x = 14

  45. Divide both sides by 7.

    7x7=147\frac{7x}{7} = \frac{14}{7} x=2x = 2

    The solution is:

    x=2\boxed{x = 2}
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