Solve the following linear equations: a) 3x = 2x + 18 b) 5t − 3 = 3t − 5 c) 5x + 9 = 5 + 3x d) 4z + 3 = 6 + 2z e) 2x − 1 = 14 − x f) 8x + 4 = 3(x − 1) + 7

$$3x - 2x = 18$$ $$x = 18$$

Check: $3(18) = 2(18) + 18 \implies 54 = 36 + 18 \implies 54 = 54$. ✅
The solution is $\boxed{x = 18}$.

$$5t - 3t = -5 + 3$$ $$2t = -2$$ $$t = \frac{-2}{2}$$ $$t = -1$$

Check: $5(-1) - 3 = 3(-1) - 5 \implies -5 - 3 = -3 - 5 \implies -8 = -8$. ✅
The solution is $\boxed{t = -1}$.

$$5x - 3x = 5 - 9$$ $$2x = -4$$ $$x = \frac{-4}{2}$$ $$x = -2$$

Check: $5(-2) + 9 = 5 + 3(-2) \implies -10 + 9 = 5 - 6 \implies -1 = -1$. ✅
The solution is $\boxed{x = -2}$.

$$4z - 2z = 6 - 3$$ $$2z = 3$$ $$z = \frac{3}{2}$$

Check: $4\left(\frac{3}{2}\right) + 3 = 6 + 2\left(\frac{3}{2}\right) \implies 6 + 3 = 6 + 3 \implies 9 = 9$. ✅
The solution is $\boxed{z = \frac{3}{2}}$.

$$2x + x = 14 + 1$$ $$3x = 15$$ $$x = \frac{15}{3}$$ $$x = 5$$

Check: $2(5) - 1 = 14 - 5 \implies 10 - 1 = 9 \implies 9 = 9$. ✅
The solution is $\boxed{x = 5}$.

$$8x + 4 = 3x - 3 + 7$$ $$8x + 4 = 3x + 4$$ $$8x - 3x = 4 - 4$$ $$5x = 0$$ $$x = \frac{0}{5}$$ $$x = 0$$

Check: $8(0) + 4 = 3(0 - 1) + 7 \implies 0 + 4 = 3(-1) + 7 \implies 4 = -3 + 7 \implies 4 = 4$. ✅
The solution is $\boxed{x = 0}$.

$$5x = 4(x + 10)$$ $$5x = 4x + 40$$ $$5x - 4x = 40$$ $$x = 40$$

Check: $40 = \frac{4}{5}(40 + 10) \implies 40 = \frac{4}{5}(50) \implies 40 = 4 \times 10 \implies 40 = 40$. ✅
The solution is $\boxed{x = 40}$.

Multiply the entire equation by 15 (the least common multiple of 3 and 15).

$$15\left(\frac{2x}{3}\right) + 15(1) = 15\left(\frac{7x}{15}\right) + 15(3)$$ $$10x + 15 = 7x + 45$$ $$10x - 7x = 45 - 15$$ $$3x = 30$$ $$x = \frac{30}{3}$$ $$x = 10$$

Check: $\frac{2(10)}{3} + 1 = \frac{7(10)}{15} + 3 \implies \frac{20}{3} + 1 = \frac{70}{15} + 3 \implies \frac{20}{3} + \frac{3}{3} = \frac{14}{3} + \frac{9}{3} \implies \frac{23}{3} = \frac{23}{3}$. ✅
The solution is $\boxed{x = 10}$.

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

Check: $2\left(\frac{7}{3}\right) + \frac{5}{3} = \frac{26}{3} - \frac{7}{3} \implies \frac{14}{3} + \frac{5}{3} = \frac{19}{3} \implies \frac{19}{3} = \frac{19}{3}$. ✅
The solution is $\boxed{y = \frac{7}{3}}$.

$$3m - 5m = -\frac{8}{5}$$ $$-2m = -\frac{8}{5}$$ $$m = \frac{-\frac{8}{5}}{-2}$$ $$m = \frac{8}{10}$$ $$m = \frac{4}{5}$$

Check: $3\left(\frac{4}{5}\right) = 5\left(\frac{4}{5}\right) - \frac{8}{5} \implies \frac{12}{5} = \frac{20}{5} - \frac{8}{5} \implies \frac{12}{5} = \frac{12}{5}$. ✅
The solution is $\boxed{m = \frac{4}{5}}$.