Collection of algebraic equations for solving linear expressions.

Here are the step-by-step solutions for each equation.

a) $10x - 1 = 15 - 6x$
1. Add $6x$ to both sides: $16x - 1 = 15$
2. Add $1$ to both sides: $16x = 16$
3. Divide by $16$: $x = 1$

b) $\frac{3x}{2} + 5 = \frac{5x}{2} - 1$
1. Multiply everything by $2$ to remove fractions: $3x + 10 = 5x - 2$
2. Subtract $3x$ from both sides: $10 = 2x - 2$
3. Add $2$ to both sides: $12 = 2x$
4. Divide by $2$: $x = 6$

c) $1\frac{1}{2}z - 2 = 3\frac{1}{4}z - 9$
1. Convert mixed numbers to improper fractions: $\frac{3}{2}z - 2 = \frac{13}{4}z - 9$
2. Multiply everything by $4$ (the common denominator): $6z - 8 = 13z - 36$
3. Subtract $6z$ from both sides: $-8 = 7z - 36$
4. Add $36$ to both sides: $28 = 7z$
5. Divide by $7$: $z = 4$

d) $9x - 8 = 11x - 10$
1. Subtract $9x$ from both sides: $-8 = 2x - 10$
2. Add $10$ to both sides: $2 = 2x$
3. Divide by $2$: $x = 1$

e) $\frac{x}{2} + \frac{x}{3} = 5$
1. Find a common denominator ($6$) and combine terms on the left: $\frac{3x}{6} + \frac{2x}{6} = 5 \rightarrow \frac{5x}{6} = 5$
2. Multiply by $6$: $5x = 30$
3. Divide by $5$: $x = 6$

f) $7 + \frac{x}{3} = 8 + \frac{x}{4}$
1. Multiply everything by $12$ (common denominator of $3$ and $4$): $84 + 4x = 96 + 3x$
2. Subtract $3x$ from both sides: $84 + x = 96$
3. Subtract $84$ from both sides: $x = 12$

g) $x - \frac{2}{3} = \frac{5x}{7} + \frac{1}{2}$
1. Multiply everything by $42$ (common denominator of $3, 7, 2$):
$42(x) - 42(\frac{2}{3}) = 42(\frac{5x}{7}) + 42(\frac{1}{2})$
$42x - 28 = 30x + 21$
2. Subtract $30x$ from both sides: $12x - 28 = 21$
3. Add $28$ to both sides: $12x = 49$
4. Divide by $12$: $x = \frac{49}{12}$ (or $4\frac{1}{12}$)

h) $2x - \frac{x}{2} + 4 = x + \frac{x}{3}$
1. Multiply everything by $6$ (common denominator of $2$ and $3$):
$12x - 3x + 24 = 6x + 2x$
$9x + 24 = 8x$
2. Subtract $9x$ from both sides: $24 = -x$
3. Multiply by $-1$: $x = -24$

i) $-\frac{17}{19}x + 51 = 0$
1. Subtract $51$ from both sides: $-\frac{17}{19}x = -51$
2. Multiply both sides by $-19$: $17x = 969$
3. Divide by $17$: $x = 57$

j) $3 - y + \frac{5y}{6} = \frac{1}{2} - \frac{y}{8}$
1. Multiply everything by $24$ (common denominator of $6, 2, 8$):
$72 - 24y + 20y = 12 - 3y$
$72 - 4y = 12 - 3y$
2. Add $4y$ to both sides: $72 = 12 + y$
3. Subtract $12$ from both sides: $y = 60$

k) $1.2 - \frac{x}{1.2} + 4.5x - \frac{x}{4.5} = 5.6 + x$
1. Convert decimals to fractions to make it easier: $1.2 = \frac{6}{5}$ and $4.5 = \frac{9}{2}$.
Equation: $\frac{6}{5} - \frac{5x}{6} + \frac{9x}{2} - \frac{2x}{9} = \frac{28}{5} + x$
2. Combine $x$ terms on the left. Common denominator for $6, 2, 9$ is $18$.
$-\frac{15x}{18} + \frac{81x}{18} - \frac{4x}{18} = \frac{62x}{18} = \frac{31x}{9}$
So: $\frac{6}{5} + \frac{31x}{9} = \frac{28}{5} + x$
3. Move constants to right, $x$ terms to left:
$\frac{31x}{9} - x = \frac{28}{5} - \frac{6}{5}$
$\frac{31x}{9} - \frac{9x}{9} = \frac{22}{5}$
$\frac{22x}{9} = \frac{22}{5}$
4. Divide both sides by $22$: $\frac{x}{9} = \frac{1}{5}$
5. Multiply by $9$: $x = \frac{9}{5}$ (or $1.8$)

l) $\frac{5}{3}(e - 6) = \frac{e}{7} + 22$
1. Expand the left side: $\frac{5e}{3} - 10 = \frac{e}{7} + 22$
2. Multiply everything by $21$ (common denominator of $3$ and $7$):
$35e - 210 = 3e + 462$
3. Subtract $3e$ from both sides: $32e - 210 = 462$
4. Add $210$ to both sides: $32e = 672$
5. Divide by $32$: $e = 21$

m) $2a - (8a + 1) - (a + 2) \times 5 = 9$
1. Expand the brackets carefully:
$2a - 8a - 1 - (5a + 10) = 9$
$2a - 8a - 1 - 5a - 10 = 9$
2. Combine like terms:
$(2 - 8 - 5)a - 11 = 9$
$-11a - 11 = 9$
3. Add $11$ to both sides: $-11a = 20$
4. Divide by $-11$: $a = -\frac{20}{11}$ (or $-1\frac{9}{11}$)

n) $2\frac{3}{5} + x = 8 \times (-4.5) - (-2x)$
1. Convert mixed number and simplify right side:
$2.6 + x = -36 + 2x$
2. Subtract $x$ from both sides: $2.6 = -36 + x$
3. Add $36$ to both sides: $x = 38.6$ (or $38\frac{3}{5}$)

o) $8\frac{1}{2}x + 2.5 = 10.7 + 2 \times 1\frac{3}{4}x$
1. Convert to decimals: $8.5x + 2.5 = 10.7 + 2(1.75)x$
2. Simplify right side: $8.5x + 2.5 = 10.7 + 3.5x$
3. Subtract $3.5x$ from both sides: $5x + 2.5 = 10.7$
4. Subtract $2.5$ from both sides: $5x = 8.2$
5. Divide by $5$: $x = 1.64$

p) $\frac{3}{8}[10(x - 5) + x] = 4x - 6\frac{1}{4}$
1. Simplify inside the square bracket first: $10x - 50 + x = 11x - 50$
Equation: $\frac{3}{8}(11x - 50) = 4x - \frac{25}{4}$
2. Multiply everything by $8$ to clear denominators:
$3(11x - 50) = 32x - 50$
$33x - 150 = 32x - 50$
3. Subtract $32x$ from both sides: $x - 150 = -50$
4. Add $150$ to both sides: $x = 100$

q) $\frac{5x}{9} - \frac{4}{15} = \frac{2x - 1}{3}$
1. Multiply everything by $45$ (common denominator of $9, 15, 3$):
$5(5x) - 3(4) = 15(2x - 1)$
$25x - 12 = 30x - 15$
2. Subtract $25x$ from both sides: $-12 = 5x - 15$
3. Add $15$ to both sides: $3 = 5x$
4. Divide by $5$: $x = \frac{3}{5}$ (or $0.6$)

r) $\frac{5}{2} - \frac{3x - 2}{0.2} = \frac{x - 0.1}{0.3}$
1. Simplify the decimal denominators by multiplying numerator and denominator by $10$:
$\frac{3x - 2}{0.2} = \frac{30x - 20}{2} = 15x - 10$
$\frac{x - 0.1}{0.3} = \frac{10x - 1}{3}$
Equation: $2.5 - (15x - 10) = \frac{10x - 1}{3}$
2. Simplify left side: $2.5 - 15x + 10 = 12.5 - 15x$
$12.5 - 15x = \frac{10x - 1}{3}$
3. Multiply by $3$: $37.5 - 45x = 10x - 1$
4. Add $45x$ to both sides: $37.5 = 55x - 1$
5. Add $1$ to both sides: $38.5 = 55x$
6. Divide by $55$: $x = \frac{38.5}{55} = 0.7$ (or $\frac{7}{10}$)

s) $-1 - 5[2x - 8(2x - 3)] = 19$
1. Work from the inside out. Expand $-8(2x - 3)$:
$-1 - 5[2x - 16x + 24] = 19$
2. Simplify inside the square bracket:
$-1 - 5[-14x + 24] = 19$
3. Expand the outer bracket:
$-1 + 70x - 120 = 19$
4. Combine constants:
$70x - 121 = 19$
5. Add $121$ to both sides: $70x = 140$
6. Divide by $70$: $x = 2$

t) $-1 - \frac{3a - a}{4} = \frac{2a - 5}{6}$
1. Simplify the numerator on the left: $3a - a = 2a$.
$-1 - \frac{2a}{4} = \frac{2a - 5}{6}$
$-1 - \frac{a}{2} = \frac{2a - 5}{6}$
2. Multiply everything by $6$ (common denominator):
$-6 - 3a = 2a - 5$
3. Add $3a$ to both sides: $-6 = 5a - 5$
4. Add $5$ to both sides: $-1 = 5a$
5. Divide by $5$: $a = -\frac{1}{5}$ (or $-0.2$)

Final Answer:
a) $x = 1$
b) $x = 6$
c) $z = 4$
d) $x = 1$
e) $x = 6$
f) $x = 12$
g) $x = \frac{49}{12}$
h) $x = -24$
i) $x = 57$
j) $y = 60$
k) $x = 1.8$
l) $e = 21$
m) $a = -\frac{20}{11}$
n) $x = 38.6$
o) $x = 1.64$
p) $x = 100$
q) $x = 0.6$
r) $x = 0.7$
s) $x = 2$
t) $a = -0.2$