Solve the following equations and check your results.

1. $3x = 2x + 18$

$3x = 2x + 18$
$\Rightarrow 3x - 2x = 18$
$\Rightarrow x = 18$

Check: $3 \times 18 = 2 \times 18 + 18$
$\Rightarrow 54 = 54$
$\Rightarrow LHS = RHS$

2. $5t - 3 = 3t - 5$

$5t - 3 = 3t - 5$
$\Rightarrow 5t - 3t = -5 + 3$
$\Rightarrow 2t = -2$
$\Rightarrow t = -1$

Check: $5 \times (-1) - 3 = 3 \times (-1) - 5$
$\Rightarrow -8 = -8$
$\Rightarrow LHS = RHS$

3. $5x + 9 = 5 + 3x$

$5x + 9 = 5 + 3x$
$\Rightarrow 5x - 3x = 5 - 9$
$\Rightarrow 2x = -4$
$\Rightarrow x = -2$

Check: $5 \times (-2) + 9 = 5 + 3 \times (-2)$
$\Rightarrow -1 = -1$
$\Rightarrow LHS = RHS$

4. $4z + 3 = 6 + 2z$

$4z + 3 = 6 + 2z$
$\Rightarrow 4z - 2z = 6 - 3$
$\Rightarrow 2z = 3$
$\Rightarrow z = \dfrac{3}{2}$

Check: $4 \times \dfrac{3}{2} + 3 = 6 + 2 \times \dfrac{3}{2}$
$\Rightarrow 9 = 9$
$\Rightarrow LHS = RHS$

5. $2x - 1 = 14 - x$

$2x - 1 = 14 - x$
$\Rightarrow 2x + x = 14 + 1$
$\Rightarrow 3x = 15$
$\Rightarrow x = 5$

Check: $2 \times 5 - 1 = 14 - 5$
$\Rightarrow 9 = 9$
$\Rightarrow LHS = RHS$

6. $8x + 4 = 3(x - 1) + 7$

$8x + 4 = 3x - 3 + 7$
$\Rightarrow 8x + 4 = 3x + 4$
$\Rightarrow 8x - 3x = 4 - 4$
$\Rightarrow 5x = 0$
$\Rightarrow x = 0$

Check: $8 \times 0 + 4 = 3 \times (0 - 1) + 7$
$\Rightarrow 4 = 4$
$\Rightarrow LHS = RHS$

7. $x = \dfrac{4}{5}(x + 10)$

$x = \dfrac{4}{5}(x + 10)$
$\Rightarrow x = \dfrac{4x}{5} + \dfrac{40}{5}$
$\Rightarrow x - \dfrac{4x}{5} = 8$
$\Rightarrow \dfrac{5x - 4x}{5} = 8$
$\Rightarrow x = 8 \times 5$
$\Rightarrow x = 40$

Check: $40 = \dfrac{4}{5}(40 + 10)$
$\Rightarrow 40 = 40$
$\Rightarrow LHS = RHS$

8. $\dfrac{2x}{3} + 1 = \dfrac{7x}{15} + 3$

$\dfrac{2x}{3} + 1 = \dfrac{7x}{15} + 3$
$\Rightarrow \dfrac{2x}{3} - \dfrac{7x}{15} = 3 - 1$
$\Rightarrow \dfrac{10x - 7x}{15} = 2$
$\Rightarrow \dfrac{3x}{15} = 2$
$\Rightarrow 3x = 30$
$\Rightarrow x = 10$

Check: $\dfrac{2 \times 10}{3} + 1 = \dfrac{7 \times 10}{15} + 3$
$\Rightarrow \dfrac{20}{3} + 1 = \dfrac{70}{15} + 3$
$\Rightarrow \dfrac{23}{3} = \dfrac{23}{3}$
$\Rightarrow LHS = RHS$

9. $2y + \dfrac{5}{3} = \dfrac{26}{3} - y$

$2y + \dfrac{5}{3} = \dfrac{26}{3} - y$
$\Rightarrow 2y + y = \dfrac{26}{3} - \dfrac{5}{3}$
$\Rightarrow 3y = \dfrac{21}{3}$
$\Rightarrow 3y = 7$
$\Rightarrow y = \dfrac{7}{3}$

Check: $2 \times \dfrac{7}{3} + \dfrac{5}{3} = \dfrac{26}{3} - \dfrac{7}{3}$
$\Rightarrow \dfrac{19}{3} = \dfrac{19}{3}$
$\Rightarrow LHS = RHS$

10. $3m = 5m - \dfrac{8}{5}$

$3m = 5m - \dfrac{8}{5}$
$\Rightarrow 5m - 3m = \dfrac{8}{5}$
$\Rightarrow 2m = \dfrac{8}{5}$
$\Rightarrow m = \dfrac{8}{5} \div 2$
$\Rightarrow m = \dfrac{4}{5}$

Check: $3 \times \dfrac{4}{5} = 5 \times \dfrac{4}{5} - \dfrac{8}{5}$
$\Rightarrow \dfrac{12}{5} = \dfrac{20}{5} - \dfrac{8}{5}$
$\Rightarrow \dfrac{12}{5} = \dfrac{12}{5}$
$\Rightarrow LHS = RHS$

1. $3x = 2x + 18$

$3x = 2x + 18$
$\Rightarrow 3x - 2x = 18$
$\Rightarrow x = 18$

जाँच: $3 \times 18 = 2 \times 18 + 18$
$\Rightarrow 54 = 54$
$\Rightarrow LHS = RHS$

2. $5t - 3 = 3t - 5$

$5t - 3 = 3t - 5$
$\Rightarrow 5t - 3t = -5 + 3$
$\Rightarrow 2t = -2$
$\Rightarrow t = -1$

जाँच: $5 \times (-1) - 3 = 3 \times (-1) - 5$
$\Rightarrow -8 = -8$
$\Rightarrow LHS = RHS$

3. $5x + 9 = 5 + 3x$

$5x + 9 = 5 + 3x$
$\Rightarrow 5x - 3x = 5 - 9$
$\Rightarrow 2x = -4$
$\Rightarrow x = -2$

जाँच: $5 \times (-2) + 9 = 5 + 3 \times (-2)$
$\Rightarrow -1 = -1$
$\Rightarrow LHS = RHS$

4. $4z + 3 = 6 + 2z$

$4z + 3 = 6 + 2z$
$\Rightarrow 4z - 2z = 6 - 3$
$\Rightarrow 2z = 3$
$\Rightarrow z = \dfrac{3}{2}$

जाँच: $4 \times \dfrac{3}{2} + 3 = 6 + 2 \times \dfrac{3}{2}$
$\Rightarrow 9 = 9$
$\Rightarrow LHS = RHS$

5. $2x - 1 = 14 - x$

$2x - 1 = 14 - x$
$\Rightarrow 2x + x = 14 + 1$
$\Rightarrow 3x = 15$
$\Rightarrow x = 5$

जाँच: $2 \times 5 - 1 = 14 - 5$
$\Rightarrow 9 = 9$
$\Rightarrow LHS = RHS$

6. $8x + 4 = 3(x - 1) + 7$

$8x + 4 = 3x - 3 + 7$
$\Rightarrow 8x + 4 = 3x + 4$
$\Rightarrow 8x - 3x = 4 - 4$
$\Rightarrow 5x = 0$
$\Rightarrow x = 0$

जाँच: $8 \times 0 + 4 = 3 \times (0 - 1) + 7$
$\Rightarrow 4 = 4$
$\Rightarrow LHS = RHS$

7. $x = \dfrac{4}{5}(x + 10)$

$x = \dfrac{4}{5}(x + 10)$
$\Rightarrow x = \dfrac{4x}{5} + \dfrac{40}{5}$
$\Rightarrow x - \dfrac{4x}{5} = 8$
$\Rightarrow \dfrac{5x - 4x}{5} = 8$
$\Rightarrow x = 8 \times 5$
$\Rightarrow x = 40$

जाँच: $40 = \dfrac{4}{5}(40 + 10)$
$\Rightarrow 40 = 40$
$\Rightarrow LHS = RHS$

8. $\dfrac{2x}{3} + 1 = \dfrac{7x}{15} + 3$

$\dfrac{2x}{3} + 1 = \dfrac{7x}{15} + 3$
$\Rightarrow \dfrac{2x}{3} - \dfrac{7x}{15} = 3 - 1$
$\Rightarrow \dfrac{10x - 7x}{15} = 2$
$\Rightarrow \dfrac{3x}{15} = 2$
$\Rightarrow 3x = 30$
$\Rightarrow x = 10$

जाँच: $\dfrac{2 \times 10}{3} + 1 = \dfrac{7 \times 10}{15} + 3$
$\Rightarrow \dfrac{20}{3} + 1 = \dfrac{70}{15} + 3$
$\Rightarrow \dfrac{23}{3} = \dfrac{23}{3}$
$\Rightarrow LHS = RHS$

9. $2y + \dfrac{5}{3} = \dfrac{26}{3} - y$

$2y + \dfrac{5}{3} = \dfrac{26}{3} - y$
$\Rightarrow 2y + y = \dfrac{26}{3} - \dfrac{5}{3}$
$\Rightarrow 3y = \dfrac{21}{3}$
$\Rightarrow 3y = 7$
$\Rightarrow y = \dfrac{7}{3}$

जाँच: $2 \times \dfrac{7}{3} + \dfrac{5}{3} = \dfrac{26}{3} - \dfrac{7}{3}$
$\Rightarrow \dfrac{19}{3} = \dfrac{19}{3}$
$\Rightarrow LHS = RHS$

10. $3m = 5m - \dfrac{8}{5}$

$3m = 5m - \dfrac{8}{5}$
$\Rightarrow 5m - 3m = \dfrac{8}{5}$
$\Rightarrow 2m = \dfrac{8}{5}$
$\Rightarrow m = \dfrac{8}{5} \div 2$
$\Rightarrow m = \dfrac{4}{5}$

जाँच: $3 \times \dfrac{4}{5} = 5 \times \dfrac{4}{5} - \dfrac{8}{5}$
$\Rightarrow \dfrac{12}{5} = \dfrac{20}{5} - \dfrac{8}{5}$
$\Rightarrow \dfrac{12}{5} = \dfrac{12}{5}$
$\Rightarrow LHS = RHS$