Linear Equation in One Variable Worksheet - Practice Problems for Solving Equations

Problem: Solve for the unknown in the following equations.

The worksheet contains 29 linear equations in one variable. Below, I will solve each equation step by step.

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#### Equations (1) to (17):

1. 8x = 6x + 10
\[
8x - 6x = 10 \implies 2x = 10 \implies x = 5
\]
Answer: \( x = 5 \)

2. 4 = 5x - 6
\[
4 + 6 = 5x \implies 10 = 5x \implies x = 2
\]
Answer: \( x = 2 \)

3. 13y = -12y + 100
\[
13y + 12y = 100 \implies 25y = 100 \implies y = 4
\]
Answer: \( y = 4 \)

4. 18x = -13x + 62
\[
18x + 13x = 62 \implies 31x = 62 \implies x = 2
\]
Answer: \( x = 2 \)

5. 5x + (-3) = 12
\[
5x - 3 = 12 \implies 5x = 15 \implies x = 3
\]
Answer: \( x = 3 \)

6. 3(x + 1) = 6
\[
3x + 3 = 6 \implies 3x = 3 \implies x = 1
\]
Answer: \( x = 1 \)

7. 7(m - 9) = 35
\[
m - 9 = 5 \implies m = 14
\]
Answer: \( m = 14 \)

8. 8(x + 3) + 2 = 42
\[
8(x + 3) = 40 \implies x + 3 = 5 \implies x = 2
\]
Answer: \( x = 2 \)

9. 16 - 3(x - 7) = -14
\[
16 - 3x + 21 = -14 \implies 37 - 3x = -14 \implies -3x = -51 \implies x = 17
\]
Answer: \( x = 17 \)

10. 3(x + 5) = 15
\[
x + 5 = 5 \implies x = 0
\]
Answer: \( x = 0 \)

11. 12(3 - x) = 48
\[
3 - x = 4 \implies -x = 1 \implies x = -1
\]
Answer: \( x = -1 \)

12. 5x + 8(2x - 9) = 54
\[
5x + 16x - 72 = 54 \implies 21x - 72 = 54 \implies 21x = 126 \implies x = 6
\]
Answer: \( x = 6 \)

13. \(\frac{x}{6} = 5\)
\[
x = 5 \times 6 \implies x = 30
\]
Answer: \( x = 30 \)

14. \(\frac{m}{3} = 4\)
\[
m = 4 \times 3 \implies m = 12
\]
Answer: \( m = 12 \)

15. \(\frac{m}{4} = \frac{1}{2}\)
\[
m = \frac{1}{2} \times 4 \implies m = 2
\]
Answer: \( m = 2 \)

16. \(\frac{x}{11} = 6\)
\[
x = 6 \times 11 \implies x = 66
\]
Answer: \( x = 66 \)

17. \(\frac{x}{5} = 7\)
\[
x = 7 \times 5 \implies x = 35
\]
Answer: \( x = 35 \)

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#### Equations (18) to (29):

18. \(\frac{x}{7} = 4.5\)
\[
x = 4.5 \times 7 \implies x = 31.5
\]
Answer: \( x = 31.5 \)

19. \(\frac{r}{9} = -11\)
\[
r = -11 \times 9 \implies r = -99
\]
Answer: \( r = -99 \)

20. \(\frac{x}{-4} = \frac{1}{8}\)
\[
x = \frac{1}{8} \times (-4) \implies x = -\frac{1}{2}
\]
Answer: \( x = -\frac{1}{2} \)

21. \(\frac{x}{-4} = \frac{3}{4}\)
\[
x = \frac{3}{4} \times (-4) \implies x = -3
\]
Answer: \( x = -3 \)

22. \(\frac{x}{2} = -39\)
\[
x = -39 \times 2 \implies x = -78
\]
Answer: \( x = -78 \)

23. \(\frac{5z - 7}{3z} = 2\)
\[
5z - 7 = 2 \cdot 3z \implies 5z - 7 = 6z \implies -7 = z \implies z = -7
\]
Answer: \( z = -7 \)

24. \(\frac{2y + 5}{y + 4} = 1\)
\[
2y + 5 = y + 4 \implies 2y - y = 4 - 5 \implies y = -1
\]
Answer: \( y = -1 \)

25. \(\frac{2x + 1}{3x - 2} = \frac{5}{9}\)
\[
9(2x + 1) = 5(3x - 2) \implies 18x + 9 = 15x - 10 \implies 18x - 15x = -10 - 9 \implies 3x = -19 \implies x = -\frac{19}{3}
\]
Answer: \( x = -\frac{19}{3} \)

26. \(\frac{2x}{3x + 1} = -3\)
\[
2x = -3(3x + 1) \implies 2x = -9x - 3 \implies 2x + 9x = -3 \implies 11x = -3 \implies x = -\frac{3}{11}
\]
Answer: \( x = -\frac{3}{11} \)

27. \(\frac{5y}{3} + \frac{2}{5} = 1\)
\[
\frac{5y}{3} = 1 - \frac{2}{5} \implies \frac{5y}{3} = \frac{5}{5} - \frac{2}{5} \implies \frac{5y}{3} = \frac{3}{5} \implies 5y = \frac{3}{5} \times 3 \implies 5y = \frac{9}{5} \implies y = \frac{9}{25}
\]
Answer: \( y = \frac{9}{25} \)

28. \(\frac{x}{2} + \frac{x}{3} + \frac{x}{4} = 13\)
\[
\text{Find the LCM of 2, 3, and 4: } \text{LCM} = 12
\]
\[
\frac{6x}{12} + \frac{4x}{12} + \frac{3x}{12} = 13 \implies \frac{13x}{12} = 13 \implies 13x = 13 \times 12 \implies 13x = 156 \implies x = 12
\]
Answer: \( x = 12 \)

29. \(\frac{2x}{3} - \frac{3x}{8} = \frac{7}{12}\)
\[
\text{Find the LCM of 3, 8, and 12: } \text{LCM} = 24
\]
\[
\frac{16x}{24} - \frac{9x}{24} = \frac{14}{24} \implies \frac{7x}{24} = \frac{14}{24} \implies 7x = 14 \implies x = 2
\]
Answer: \( x = 2 \)

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Final Answers:

\[
\boxed{
\begin{array}{ll}
(1) & x = 5 \\
(2) & x = 2 \\
(3) & y = 4 \\
(4) & x = 2 \\
(5) & x = 3 \\
(6) & x = 1 \\
(7) & m = 14 \\
(8) & x = 2 \\
(9) & x = 17 \\
(10) & x = 0 \\
(11) & x = -1 \\
(12) & x = 6 \\
(13) & x = 30 \\
(14) & m = 12 \\
(15) & m = 2 \\
(16) & x = 66 \\
(17) & x = 35 \\
(18) & x = 31.5 \\
(19) & r = -99 \\
(20) & x = -\frac{1}{2} \\
(21) & x = -3 \\
(22) & x = -78 \\
(23) & z = -7 \\
(24) & y = -1 \\
(25) & x = -\frac{19}{3} \\
(26) & x = -\frac{3}{11} \\
(27) & y = \frac{9}{25} \\
(28) & x = 12 \\
(29) & x = 2 \\
\end{array}
}
\]