0.7e: Exercises - Linear Equations

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A: Check a Solution

Exercise \(\PageIndex{1}\)

\( \bigstar \) Determine whether or not the given value is a solution.

\(−5x + 4 = −1 ; x = −1\)
\(4x − 3 = −7 ; x = −1\)
\(3y − 4 = 5; y = \frac{9}{3}\)

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

\(ax − b = 0; x = \dfrac{b}{a}\)
\(ax + b = 2b; x = \dfrac{b}{a}\)

Answers to odd exercises:: 1. No \(\qquad\) 3. Yes \(\qquad\) 5. No \(\qquad\) 7. Yes

B: Solve Linear Equations (I)

Exercise \(\PageIndex{2}\)

\( \bigstar \) Solve.

\(5x − 3 = 27\)
\(6x − 7 = 47\)
\(4x + 13 = 35\)
\(6x − 9 = 18\)
\(9a + 10 = 10\)
\(5 − 3a = 5\)

\(−8t + 5 = 15\)
\(−9t + 12 = 33\)
\(\dfrac{2}{3} x + \dfrac{1}{2} = 1\)
\(\dfrac{3}{8} x + \dfrac{5}{4} = \dfrac{3}{2}\)
\(\dfrac{1 − 3y}{5} = 2\)

\(\dfrac{2 − 5y}{6} = −8\)
\(7 − y = 22\)
\(6 − y = 12\)
Solve for \(x: ax − b = c\)
Solve for \(x: ax + b = 0\)

Answers to odd exercises:: 11. \(6\) \(\qquad\) 13. \(\frac{11}{2}\) \(\qquad\) 15. \(0\) \(\qquad\) 17. \(−\frac{5}{4}\) \(\qquad\) 19. \(\frac{3}{4}\) \(\qquad\) 21. \(−3\) \(\qquad\) 23. \(−15\) \(\qquad\) 25. \(x = \frac{b+c}{a}\)

C: Solve Linear Equations (II)

Exercise \(\PageIndex{3}\)

\( \bigstar \) Solve.

\(6x − 5 + 2x = 19\)
\(7 − 2x + 9 = 24\)
\(12x − 2 − 9x = 5x + 8\)
\(16 − 3x − 22 = 8 − 4x\)
\(5y − 6 − 9y = 3 − 2y + 8\)
\(7 − 9y + 12 = 3y + 11 − 11y\)
\(3 + 3a − 11 = 5a − 8 − 2a\)
\(2 − 3a = 5a + 7 − 8a\)
\(\dfrac{1}{3} x −\dfrac{3}{2} + \dfrac{5}{2} x = \dfrac{5}{6} x + \dfrac{1}{4}\)
\(\dfrac{5}{8} + \dfrac{1}{5} x −\dfrac{3}{4} = \dfrac{3}{10} x − \dfrac{1}{4}\)
\(1.2x − 0.5 − 2.6x = 2 − 2.4x\)
\(1.59 − 3.87x = 3.48 − 4.1x − 0.51\)
\(5 − 10x = 2x + 8 − 12x\)

\(8x − 3 − 3x = 5x − 3\)
\(5 (y + 2) = 3 (2y − 1) + 10\)
\(7 (y − 3) = 4 (2y + 1) − 21\)
\(7 − 5 (3t − 9) = 22\)
\(10 − 5 (3t + 7) = 20\)
\(5 − 2x = 4 − 2 (x − 4)\)
\(2 (4x − 5) + 7x = 5 (3x − 2)\)
\(4 (4a − 1) = 5 (a − 3) + 2 (a − 2)\)
\(6 (2b − 1) + 24b = 8 (3b − 1)\)
\(\dfrac{2}{3} (x + 18) + 2 = \dfrac{1}{3} x − 13\)
\(\dfrac{2}{5} x − \dfrac{1}{2} (6x − 3) = \dfrac{4}{3}\)
\(1.2 (2x + 1) + 0.6x = 4x\)
\(6 + 0.5 (7x − 5) = 2.5x + 0.3\)

\(5 (y + 3) = 15 (y + 1) − 10y\)
\(3 (4 − y) − 2 (y + 7) = −5y\)
\(\dfrac{1}{5} (2a + 3) −\dfrac{1}{2} = \dfrac{1}{3} a + \dfrac{1}{10}\)
\(\dfrac{3}{2} a = \dfrac{3}{4} (1 + 2a) −\dfrac{1}{5} (a + 5)\)
\(6 − 3 (7x + 1) = 7 (4 − 3x)\)
\(6 (x − 6) − 3 (2x − 9) = −9\)
\(\dfrac{3}{4} (y − 2) + \dfrac{2}{3} (2y + 3) = 3\)
\(\dfrac{5}{4} − \dfrac{1}{2} (4y − 3) = \dfrac{2}{5} (y − 1)\)
\(−2 (3x + 1) − (x − 3) = −7x + 1\)
\(6 (2x + 1) − (10x + 9) = 0\)

Answers to odd exercises:: 31. \(3\) \(\qquad\) 33. \(−5\) \(\qquad\) 35. \(−\frac{17}{2}\) \(\qquad\) 37. \(ℝ\) \(\qquad\) 39. \(\frac{7}{8}\) \(\qquad\) 41. \(2.5\) \(\qquad\) 43. \(Ø\) \(\qquad\) 45. \(3\) \(\qquad\) 47. \(2\) \(\qquad\) 49. \(Ø\)
51. \(−\frac{5}{3}\) \(\qquad\) 53. \(−81\) \(\qquad\) 55. \(1.2\) \(\qquad\) 57. \(ℝ\) \(\qquad\) 59. \(0\) \(\qquad\) 61. \(Ø\) \(\qquad\) 63. \(\frac{6}{5}\) \(\qquad\) 65. \(ℝ\) \(\qquad\)

C: Solve Linear Formulas

Exercise \(\PageIndex{4}\)

\( \bigstar \) Solve.

Solve for \(w: P = 2l + 2w\)
Solve for \(a: P = a + b + c\)
Solve for \(t: D = rt\)
Solve for \(w: V = lwh\)

Solve for \(b: A = \dfrac{1}{2} bh\)
Solve for \(a:s = \dfrac{1}{2}at^{2}\)
Solve for \(a: A = \dfrac{1}{2}h (a + b)\)

Solve for \(h: V = \dfrac{1}{3}πr^{2}h\)
Solve for \(F: C = \dfrac{5}{9} (F − 32)\)
Solve for \(x: ax + b = c\)

Answers to odd exercises:: 67. \(w = \frac{P − 2l}{2}\) \(\qquad\) 69. \(t = \frac{D}{r}\) \(\qquad\) 71. \(b = \frac{2A}{h}\) \(\qquad\) 73. \(a = \frac{2A}{h} − b\) \(\qquad\) 75. \(F = \frac{9}{5} C + 32\).