Solving Linear Equations (D) Worksheet | Fun and Engaging PDF ... - Free Printable

Here are the solutions for the linear equations on the worksheet. I have worked through each one step-by-step to ensure accuracy.

Section A

1) $\frac{2x + 5}{3} = 11$
* Multiply by 3: $2x + 5 = 33$
* Subtract 5: $2x = 28$
* Divide by 2: $x = 14$

2) $\frac{8 - 3x}{2} = 5$
* Multiply by 2: $8 - 3x = 10$
* Subtract 8: $-3x = 2$
* Divide by -3: $x = -\frac{2}{3}$

3) $\frac{5 - 9x}{6} = -2$
* Multiply by 6: $5 - 9x = -12$
* Subtract 5: $-9x = -17$
* Divide by -9: $x = \frac{17}{9}$ (or $1 \frac{8}{9}$)

4) $\frac{7x + 6}{3} - 9 = -12$
* Add 9: $\frac{7x + 6}{3} = -3$
* Multiply by 3: $7x + 6 = -9$
* Subtract 6: $7x = -15$
* Divide by 7: $x = -\frac{15}{7}$ (or $-2 \frac{1}{7}$)

5) $8x + \frac{1 - 4x}{8} = 7$
* Multiply everything by 8: $64x + (1 - 4x) = 56$
* Simplify: $60x + 1 = 56$
* Subtract 1: $60x = 55$
* Divide by 60: $x = \frac{55}{60}$ which simplifies to $x = \frac{11}{12}$

6) $\frac{5}{x} = -6$
* Multiply by $x$: $5 = -6x$
* Divide by -6: $x = -\frac{5}{6}$

7) $\frac{11}{4x} + 9 = 3$
* Subtract 9: $\frac{11}{4x} = -6$
* Multiply by $4x$: $11 = -24x$
* Divide by -24: $x = -\frac{11}{24}$

8) $5 - \frac{3x}{4} = 8x$
* Multiply by 4: $20 - 3x = 32x$
* Add $3x$: $20 = 35x$
* Divide by 35: $x = \frac{20}{35}$ which simplifies to $x = \frac{4}{7}$

9) $2 + \frac{4x}{3} - 7 = 1$
* Combine numbers ($2 - 7 = -5$): $\frac{4x}{3} - 5 = 1$
* Add 5: $\frac{4x}{3} = 6$
* Multiply by 3: $4x = 18$
* Divide by 4: $x = \frac{18}{4}$ which simplifies to $x = \frac{9}{2}$ (or $4.5$)

10) $4 - \frac{3x}{2} = 3x + 5$
* Multiply by 2: $8 - 3x = 6x + 10$
* Add $3x$: $8 = 9x + 10$
* Subtract 10: $-2 = 9x$
* Divide by 9: $x = -\frac{2}{9}$

11) $6 - \frac{2}{x} = 10$
* Subtract 6: $-\frac{2}{x} = 4$
* Multiply by $x$: $-2 = 4x$
* Divide by 4: $x = -\frac{2}{4}$ which simplifies to $x = -\frac{1}{2}$ (or $-0.5$)

12) $4 - \frac{2x}{9} + x = -1$
* Multiply by 9: $36 - 2x + 9x = -9$
* Simplify: $36 + 7x = -9$
* Subtract 36: $7x = -45$
* Divide by 7: $x = -\frac{45}{7}$ (or $-6 \frac{3}{7}$)

---

Section B

1) $4(2x - 3) = 8(2x + 5)$
* Expand: $8x - 12 = 16x + 40$
* Subtract $8x$: $-12 = 8x + 40$
* Subtract 40: $-52 = 8x$
* Divide by 8: $x = -\frac{52}{8}$ which simplifies to $x = -\frac{13}{2}$ (or $-6.5$)

2) $3(4x - 5) = 5(2x - 5)$
* Expand: $12x - 15 = 10x - 25$
* Subtract $10x$: $2x - 15 = -25$
* Add 15: $2x = -10$
* Divide by 2: $x = -5$

3) $8(6x + 2) = 5(x - 2)$
* Expand: $48x + 16 = 5x - 10$
* Subtract $5x$: $43x + 16 = -10$
* Subtract 16: $43x = -26$
* Divide by 43: $x = -\frac{26}{43}$

4) $2(3x - 4) = 7(11 - 2x)$
* Expand: $6x - 8 = 77 - 14x$
* Add $14x$: $20x - 8 = 77$
* Add 8: $20x = 85$
* Divide by 20: $x = \frac{85}{20}$ which simplifies to $x = \frac{17}{4}$ (or $4.25$)

5) $7(5 - x) = -4(x - 11)$
* Expand: $35 - 7x = -4x + 44$
* Add $7x$: $35 = 3x + 44$
* Subtract 44: $-9 = 3x$
* Divide by 3: $x = -3$

6) $-4(x - 8) = -6(4 + 3x)$
* Expand: $-4x + 32 = -24 - 18x$
* Add $18x$: $14x + 32 = -24$
* Subtract 32: $14x = -56$
* Divide by 14: $x = -4$

7) $7(4 - 3x) = 2(8x - 9) + 6$
* Expand: $28 - 21x = 16x - 18 + 6$
* Simplify right side: $28 - 21x = 16x - 12$
* Add $21x$: $28 = 37x - 12$
* Add 12: $40 = 37x$
* Divide by 37: $x = \frac{40}{37}$ (or $1 \frac{3}{37}$)

8) $-6(3 - 4x) + 2x = 8(x + 11)$
* Expand: $-18 + 24x + 2x = 8x + 88$
* Simplify left side: $-18 + 26x = 8x + 88$
* Subtract $8x$: $-18 + 18x = 88$
* Add 18: $18x = 106$
* Divide by 18: $x = \frac{106}{18}$ which simplifies to $x = \frac{53}{9}$ (or $5 \frac{8}{9}$)

9) $3(2x - 6) = 3 - 4(3 - x)$
* Expand: $6x - 18 = 3 - 12 + 4x$
* Simplify right side: $6x - 18 = -9 + 4x$
* Subtract $4x$: $2x - 18 = -9$
* Add 18: $2x = 9$
* Divide by 2: $x = \frac{9}{2}$ (or $4.5$)

10) $9(2x - 1) - 3x = 3(12 + x)$
* Expand: $18x - 9 - 3x = 36 + 3x$
* Simplify left side: $15x - 9 = 36 + 3x$
* Subtract $3x$: $12x - 9 = 36$
* Add 9: $12x = 45$
* Divide by 12: $x = \frac{45}{12}$ which simplifies to $x = \frac{15}{4}$ (or $3.75$)

11) $4x - (2x - 8) = 5(1 + 2x)$
* Expand (watch the negative sign): $4x - 2x + 8 = 5 + 10x$
* Simplify left side: $2x + 8 = 5 + 10x$
* Subtract $2x$: $8 = 5 + 8x$
* Subtract 5: $3 = 8x$
* Divide by 8: $x = \frac{3}{8}$ (or $0.375$)

12) $10 - 6(8x - 2) = 9x - (3 + 4x)$
* Expand: $10 - 48x + 12 = 9x - 3 - 4x$
* Simplify both sides: $22 - 48x = 5x - 3$
* Add $48x$: $22 = 53x - 3$
* Add 3: $25 = 53x$
* Divide by 53: $x = \frac{25}{53}$

---

Section C

1) $\frac{5x - 2}{3} = \frac{4x + 1}{2}$
* Cross multiply: $2(5x - 2) = 3(4x + 1)$
* Expand: $10x - 4 = 12x + 3$
* Subtract $10x$: $-4 = 2x + 3$
* Subtract 3: $-7 = 2x$
* Divide by 2: $x = -\frac{7}{2}$ (or $-3.5$)

2) $\frac{7x - 8}{5} = \frac{2x + 5}{4}$
* Cross multiply: $4(7x - 8) = 5(2x + 5)$
* Expand: $28x - 32 = 10x + 25$
* Subtract $10x$: $18x - 32 = 25$
* Add 32: $18x = 57$
* Divide by 18: $x = \frac{57}{18}$ which simplifies to $x = \frac{19}{6}$ (or $3 \frac{1}{6}$)

3) $\frac{-8x - 1}{2} = \frac{5 - 3x}{6}$
* Cross multiply: $6(-8x - 1) = 2(5 - 3x)$
* Expand: $-48x - 6 = 10 - 6x$
* Add $48x$: $-6 = 10 + 42x$
* Subtract 10: $-16 = 42x$
* Divide by 42: $x = -\frac{16}{42}$ which simplifies to $x = -\frac{8}{21}$

4) $\frac{5(x + 11)}{3} = \frac{3(1 + x)}{2}$
* Cross multiply: $2[5(x + 11)] = 3[3(1 + x)]$
* Expand inside brackets first or distribute constants: $10(x + 11) = 9(1 + x)$
* Expand: $10x + 110 = 9 + 9x$
* Subtract $9x$: $x + 110 = 9$
* Subtract 110: $x = -101$

5) $\frac{3(2 + 5x)}{4} = \frac{2(6x - 3)}{5}$
* Cross multiply: $5[3(2 + 5x)] = 4[2(6x - 3)]$
* Simplify coefficients: $15(2 + 5x) = 8(6x - 3)$
* Expand: $30 + 75x = 48x - 24$
* Subtract $48x$: $30 + 27x = -24$
* Subtract 30: $27x = -54$
* Divide by 27: $x = -2$

6) $\frac{2(3x - 5)}{3} = \frac{-4(x - 2)}{7}$
* Cross multiply: $7[2(3x - 5)] = 3[-4(x - 2)]$
* Simplify coefficients: $14(3x - 5) = -12(x - 2)$
* Expand: $42x - 70 = -12x + 24$
* Add $12x$: $54x - 70 = 24$
* Add 70: $54x = 94$
* Divide by 54: $x = \frac{94}{54}$ which simplifies to $x = \frac{47}{27}$ (or $1 \frac{20}{27}$)

7) $\frac{1}{2}(2x - 6) = \frac{1}{4}(8 - 12x)$
* Multiply by 4 to clear denominators: $2(2x - 6) = 1(8 - 12x)$
* Expand: $4x - 12 = 8 - 12x$
* Add $12x$: $16x - 12 = 8$
* Add 12: $16x = 20$
* Divide by 16: $x = \frac{20}{16}$ which simplifies to $x = \frac{5}{4}$ (or $1.25$)

8) $\frac{1}{2}(5x + 7) = \frac{3}{4}(3x - 1)$
* Multiply by 4 to clear denominators: $2(5x + 7) = 3(3x - 1)$
* Expand: $10x + 14 = 9x - 3$
* Subtract $9x$: $x + 14 = -3$
* Subtract 14: $x = -17$

9) $\frac{5}{3x + 1} = 12$
* Multiply by $(3x + 1)$: $5 = 12(3x + 1)$
* Expand: $5 = 36x + 12$
* Subtract 12: $-7 = 36x$
* Divide by 36: $x = -\frac{7}{36}$

10) $\frac{x + 2}{x + 3} = 4$
* Multiply by $(x + 3)$: $x + 2 = 4(x + 3)$
* Expand: $x + 2 = 4x + 12$
* Subtract $x$: $2 = 3x + 12$
* Subtract 12: $-10 = 3x$
* Divide by 3: $x = -\frac{10}{3}$ (or $-3 \frac{1}{3}$)

11) $\frac{2x - 9}{3x - 2} = -3$
* Multiply by $(3x - 2)$: $2x - 9 = -3(3x - 2)$
* Expand: $2x - 9 = -9x + 6$
* Add $9x$: $11x - 9 = 6$
* Add 9: $11x = 15$
* Divide by 11: $x = \frac{15}{11}$ (or $1 \frac{4}{11}$)

12) $\frac{2}{3x + 10} = \frac{1}{x - 1}$
* Cross multiply: $2(x - 1) = 1(3x + 10)$
* Expand: $2x - 2 = 3x + 10$
* Subtract $2x$: $-2 = x + 10$
* Subtract 10: $x = -12$

13) $\frac{2}{7x + 3} = \frac{9}{2x - 5}$
* Cross multiply: $2(2x - 5) = 9(7x + 3)$
* Expand: $4x - 10 = 63x + 27$
* Subtract $4x$: $-10 = 59x + 27$
* Subtract 27: $-37 = 59x$
* Divide by 59: $x = -\frac{37}{59}$

14) $\frac{8}{6x + 12} = -\frac{11}{7x - 10}$
* Cross multiply: $8(7x - 10) = -11(6x + 12)$
* Expand: $56x - 80 = -66x - 132$
* Add $66x$: $122x - 80 = -132$
* Add 80: $122x = -52$
* Divide by 122: $x = -\frac{52}{122}$ which simplifies to $x = -\frac{26}{61}$

Final Answer:
Section A
1) 14
2) -2/3
3) 17/9
4) -15/7
5) 11/12
6) -5/6
7) -11/24
8) 4/7
9) 9/2 (or 4.5)
10) -2/9
11) -1/2 (or -0.5)
12) -45/7

Section B
1) -13/2 (or -6.5)
2) -5
3) -26/43
4) 17/4 (or 4.25)
5) -3
6) -4
7) 40/37
8) 53/9
9) 9/2 (or 4.5)
10) 15/4 (or 3.75)
11) 3/8 (or 0.375)
12) 25/53

Section C
1) -7/2 (or -3.5)
2) 19/6
3) -8/21
4) -101
5) -2
6) 47/27
7) 5/4 (or 1.25)
8) -17
9) -7/36
10) -10/3
11) 15/11
12) -12
13) -37/59
14) -26/61