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1.1E: Exercises - Solving Linear Equations in One Variable

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    Exercise \(\PageIndex{1}\)

    Determine whether or not the given value is a solution.

    1. \(−5x + 4 = −1 ; x = −1\)
    2. \(4x − 3 = −7 ; x = −1\)
    3. \(3y − 4 = 5; y = \frac{9}{3}\)
    4. \(−2y + 7 = 12 ; y = −\frac{5}{2}\)
    5. \(3a − 6 = 18 − a; a = −3\)
    6. \(5 (2t − 1) = 2 − t; t = 2\)
    Answer

    1. No

    3. Yes

    5. No

    Exercise \(\PageIndex{2}\)

    Solve.

    1. \(5x − 3 = 27\)
    2. \(6x − 7 = 47\)
    3. \(4x + 13 = 35\)
    4. \(6x − 9 = 18\)
    5. \(9a + 10 = 10\)
    6. \(5 − 3a = 5\)
    7. \(−8t + 5 = 15\)
    8. \(−9t + 12 = 33\)
    9. \(\frac{2}{3} x + \frac{1}{2} = 1\)
    10. \(\frac{3}{8} x + \frac{5}{4} = \frac{3}{2}\)
    11. \(\frac{1 − 3y}{5} = 2\)
    12. \(\frac{2 − 5y}{6} = −8\)
    13. \(7 − y = 22\)
    14. \(6 − y = 12\)
    Answer

    1. \(6\)

    3. \(\frac{11}{2}\)

    5. \(0\)

    7. \(−\frac{5}{4}\)

    9. \(\frac{3}{4}\)

    11. \(−3\)

    13. \(−15\)

    Exercise \(\PageIndex{3}\)

    Solve.

    1. \(6x − 5 + 2x = 19\)
    2. \(7 − 2x + 9 = 24\)
    3. \(12x − 2 − 9x = 5x + 8\)
    4. \(16 − 3x − 22 = 8 − 4x\)
    5. \(5y − 6 − 9y = 3 − 2y + 8\)
    6. \(7 − 9y + 12 = 3y + 11 − 11y\)
    7. \(3 + 3a − 11 = 5a − 8 − 2a\)
    8. \(2 − 3a = 5a + 7 − 8a\)
    9. \(\frac{1}{3} x −\frac{3}{2} + \frac{5}{2} x = \frac{5}{6} x + \frac{1}{4}\)
    10. \(\frac{5}{8} + \frac{1}{5} x −\frac{3}{4} = \frac{3}{10} x − \frac{1}{4}\)
    11. \(1.2x − 0.5 − 2.6x = 2 − 2.4x\)
    12. \(1.59 − 3.87x = 3.48 − 4.1x − 0.51\)
    13. \(5 − 10x = 2x + 8 − 12x\)
    14. \(8x − 3 − 3x = 5x − 3\)
    15. \(5 (y + 2) = 3 (2y − 1) + 10\)
    16. \(7 (y − 3) = 4 (2y + 1) − 21\)
    17. \(7 − 5 (3t − 9) = 22\)
    18. \(10 − 5 (3t + 7) = 20\)
    19. \(5 − 2x = 4 − 2 (x − 4)\)
    20. \(2 (4x − 5) + 7x = 5 (3x − 2)\)
    21. \(4 (4a − 1) = 5 (a − 3) + 2 (a − 2)\)
    22. \(6 (2b − 1) + 24b = 8 (3b − 1)\)
    23. \(\frac{2}{3} (x + 18) + 2 = \frac{1}{3} x − 13\)
    24. \(\frac{2}{5} x − \frac{1}{2} (6x − 3) = \frac{4}{3}\)
    25. \(1.2 (2x + 1) + 0.6x = 4x\)
    26. \(6 + 0.5 (7x − 5) = 2.5x + 0.3\)
    27. \(5 (y + 3) = 15 (y + 1) − 10y\)
    28. \(3 (4 − y) − 2 (y + 7) = −5y\)
    29. \(\frac{1}{5} (2a + 3) −\frac{1}{2} = \frac{1}{3} a + \frac{1}{10}\)
    30. \(\frac{3}{2} a = \frac{3}{4} (1 + 2a) −\frac{1}{5} (a + 5)\)
    31. \(6 − 3 (7x + 1) = 7 (4 − 3x)\)
    32. \(6 (x − 6) − 3 (2x − 9) = −9\)
    33. \(\frac{3}{4} (y − 2) + \frac{2}{3} (2y + 3) = 3\)
    34. \(\frac{5}{4} − \frac{1}{2} (4y − 3) = \frac{2}{5} (y − 1)\)
    35. \(−2 (3x + 1) − (x − 3) = −7x + 1\)
    36. \(6 (2x + 1) − (10x + 9) = 0\)
    Answer

    1. \(3\)

    3. \(−5\)

    5. \(−\frac{17}{2}\)

    7. \(ℝ\)

    9. \(\frac{7}{8}\)

    11. \(2.5\)

    13. \(Ø\)

    15. \(3\)

    17. \(2\)

    19. \(Ø\)

    21. \(−\frac{5}{3}\)

    23. \(−81\)

    25. \(1.2\)

    27. \(ℝ\)

    29. \(0\)

    31. \(Ø\)

    33. \(\frac{6}{5}\)

    35. \(ℝ\)

    Exercise \(\PageIndex{4}\)

    Set up an algebraic equation then solve.

    Number Problems

    1. When \(3\) is subtracted from the sum of a number and \(10\) the result is \(2\). Find the number.
    2. The sum of \(3\) times a number and \(12\) is equal to \(3\). Find the number.
    3. Three times the sum of a number and \(6\) is equal to \(5\) times the number. Find the number.
    4. Twice the sum of a number and \(4\) is equal to \(3\) times the sum of the number and \(1\). Find the number.
    5. A larger integer is \(1\) more than \(3\) times another integer. If the sum of the integers is \(57\), find the integers.
    6. A larger integer is \(5\) more than twice another integer. If the sum of the integers is \(83\), find the integers.
    7. One integer is \(3\) less than twice another integer. Find the integers if their sum is \(135\).
    8. One integer is \(10\) less than \(4\) times another integer. Find the integers if their sum is \(100\).
    9. The sum of three consecutive integers is \(339\). Find the integers.
    10. The sum of four consecutive integers is \(130\). Find the integers.
    11. The sum of three consecutive even integers is \(174\). Find the integers.
    12. The sum of four consecutive even integers is \(116\). Find the integers.
    13. The sum of three consecutive odd integers is \(81\). Find the integers.
    14. The sum of four consecutive odd integers is \(176\). Find the integers.
    Answer

    1. \(−5\)

    3. \(9\)

    5. \(14, 43\)

    7. \(46, 89\)

    9. \(112, 113, 114\)

    11. \(56, 58, 60\)

    13. \(25, 27, 29\)

    Exercise \(\PageIndex{5}\)

    Geometry Problems

    1. The length of a rectangle is \(5\) centimeters less than twice its width. If the perimeter is \(134\) centimeters, find the length and width.
    2. The length of a rectangle is \(4\) centimeters more than \(3\) times its width. If the perimeter is \(64\) centimeters, find the length and width.
    3. The width of a rectangle is one-half that of its length. If the perimeter measures \(36\) inches, find the dimensions of the rectangle.
    4. The width of a rectangle is \(4\) inches less than its length. If the perimeter measures \(72\) inches, find the dimensions of the rectangle.
    5. The perimeter of a square is \(48\) inches. Find the length of each side.
    6. The perimeter of an equilateral triangle is \(96\) inches. Find the length of each side.
    7. The circumference of a circle measures \(80π\) units. Find the radius.
    8. The circumference of a circle measures \(25\) centimeters. Find the radius rounded off to the nearest hundredth.
    Answer

    1. Width: \(24\) centimeters; length: \(43\) centimeters

    3. Width: \(6\) inches; length: \(12\) inches

    5. \(12\) inches

    7. \(40\) units

    Footnotes

    138Linear expressions related with the symbols \(≤, <, ≥,\) and \(>\).

    139A real number that produces a true statement when its value is substituted for the variable.

    140Properties used to obtain equivalent inequalities and used as a means to solve them.

    141Inequalities that share the same solution set.

    142Two or more inequalities in one statement joined by the word “and” or by the word “or.”

    1.1E: Exercises - Solving Linear Equations in One Variable is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.