Question

please solve all these equations

Answer 1

Step-by-step explanation:

(b) $\dfrac{2}{6x - 19} = \dfrac{3}{2x - 11}$

Cross Multiplying, we get

→ 2(2x - 11) = 3(6x - 19)

→ 4x - 22 = 18x - 57

→ 4x - 18x = - 57 + 22

→ - 14x = - 35

→ 14x = 35

→ x = 35/14

(c) $\dfrac{5(2x - 1) + 3(4x + 3)}{4x - 1} = 5$

→ $\dfrac{10x - 5 + 12x + 9}{4x - 1} = 5$

→ $\dfrac{22x + 4}{4x - 1} = 5$

Cross Multiplying, we get

→ 22x + 4 = 5(4x - 1)

→ 22x + 4 = 20x - 5

→ 22x - 20x = - 5 - 4

→ 2x = - 9

→ x = - 9/2

(d) $\dfrac{2(3x - 1) - (2x + 1)}{7x - 2} = \dfrac{1}{2}$

→ $\dfrac{6x - 2 - 2x - 1}{7x - 2} = \dfrac{1}{2}$

→ $\dfrac{4x - 3}{7x- 2} = \dfrac{1}{2}$

Cross Multiplying, we get

→ 2(4x - 3) = 1(7x - 2)

→ 8x - 6 = 7x - 2

→ 8x - 7x = - 2 + 6

→ x = 4

(e) $\dfrac{x + 5}{2} + \dfrac{x - 5}{3} = \dfrac{25}{6}$

→ $\dfrac{3(x + 5) + 2(x - 5)}{6} = \dfrac{25}{6}$

→ 3(x + 5) + 2(x - 5) = 25

→ 3x + 15 + 2x - 10 = 25

→ 5x + 5 = 25

→ 5x = 25 - 5

→ 5x = 20

→ x = 20/5

→ x = 4

(f) (5x - 1)(x + 3) = (x - 5)(5x + 1) + 40

→ 5x(x + 3) - 1(x + 3) = x(5x + 1) - 5(5x + 1) + 40

→ 5x² + 15x - x - 3 = 5x² + x - 25x - 5 + 40

→ 15x - x - 3 = x - 25x - 5 + 40

→ 14x - 3 = - 24x + 35

→ 14x + 24x = 35 + 3

→ 38x = 38

→ x = 38/38

→ x = 1

(g) $10 - (\frac{x - 1}{2}) - (\frac{x - 2}{3}) = \frac{x - 3}{4}$

→ $10 = \dfrac{x - 3}{4} + \dfrac{x - 1}{2} + \dfrac{x - 2}{3}$

→ $10 = \dfrac{3(x - 3) + 6(x - 1) + 4(x - 2)}{12}$

→ (10)(12) = 3(x - 3) + 6(x - 1) + 4(x - 2)

→ 120 = 3x - 9 + 6x - 6 + 4x - 8

→ 120 = 13x - 23

→ 13x = 120 + 23

→ 13x = 143

→ x = 143/13

→ x = 11

(h) $\dfrac{0.5(x - 0.4)}{0.35} - \dfrac{0.6(x - 2.71)}{0.42} =x + 6.1$

→ $\dfrac{0.5x - 0.20}{0.35} - \dfrac{0.6x - 1.626}{0.42} = x + 6.1$

→ $\dfrac{6(0.5x - 0.20) - 5(0.6x - 1.626)}{2.10} = x + 6.1$

→ $\dfrac{3.0x - 1.20 - 3.0x + 8.13}{2.10} = x + 6.1$

→ $\dfrac{ - 1.20 + 8.13}{2.10} = x + 6.1$

→ $\dfrac{6.93}{2.10} = x + 6.1$

→ $\dfrac{693}{210} = x + 6.1$

→ 3.3 = x + 6.1

→ x = 3.3 - 6.1

→ x = - 2.8

(i) (x + 4)² - (x - 5)² = 9

Identity : (a + b)² = a² + b² + 2ab

Here, a = x, b = 4

(a - b)² = a² + b² - 2ab

Here, a = x, b = 5

→ [ (x)² + (4)² + 2(x)(4) ] - [ (x)² + (5)² - 2(x)(5) ] = 9

→ [ x² + 16 + 8x ] - [ x² + 25 - 10x ] = 9

→ x² + 16 + 8x - x² - 25 + 10x = 9

→ 16 + 8x - 25 + 10x = 9

→ - 9 + 18x = 9

→ 18x = 9 + 9

→ 18x = 18

→ x = 18/18

→ x = 1

(j) $\dfrac{7x - 1}{4} - \dfrac{1}{7} (2x - \frac{1 - x}{2} ) = 4$

→ $\frac{7x - 1}{4} - \dfrac{1}{7} ( \frac{2(2x) - (1 - x)}{2} ) = 4$

→ $\dfrac{7x - 1}{4} - \dfrac{1}{7} ( \frac{4x - 1 + x}{2} ) = 4$

→ $\dfrac{7x - 1}{4} - \dfrac{1}{7} ( \frac{5x - 1}{2} ) = 4$

→ $\dfrac{7x - 1}{4} - \dfrac{5x - 1}{14} = 4$

→ $\dfrac{7(7x - 1) - 2(5x - 1)}{28} = 4$

→ $\dfrac{49x - 7 - 10x + 2}{28} = 4$

→ $\dfrac{39x - 5}{28} = 4$

→ 39x - 5 = (4)(28)

→ 39x - 5 = 112

→ 39x = 112 + 5

→ 39x = 117

→ x = 117/39

→ x = 3

(k) $\dfrac{3x + 5}{4x + 2} = \dfrac{3x + 4}{4x + 7}$

Cross Multiplying, we get

→ (3x + 5)(4x + 7) = (3x + 4)(4x + 2)

→ 3x(4x + 7) + 5(4x + 7) = 3x(4x + 2) + 4(4x + 2)

→ 12x² + 21x + 20x + 35 = 12x² + 6x + 16x + 8

→ 21x + 20x + 35 = 6x + 16x + 8

→ 41x + 35 = 22x + 8

→ 41x - 22x = 8 - 35

→ 19x = - 27

→ x = - 27/19

(l) $\dfrac{0.8 + 0.7x}{x} = 0.86$

Cross Multiplying, we get

→ 0.8 + 0.7x = (0.86)(x)

→ 0.8 + 0.7x = 0.86x

→ 0.86x - 0.7x = 0.8

→ 0.16x = 0.8

→ x = 0.8/0.16

→ x = 80/16

→ x = 5

Answer 2

(b) \dfrac{2}{6x - 19} = \dfrac{3}{2x - 11}

6x−19

2

=

2x−11

3

Cross Multiplying, we get

→ 2(2x - 11) = 3(6x - 19)

→ 4x - 22 = 18x - 57

→ 4x - 18x = - 57 + 22

→ - 14x = - 35

→ 14x = 35

→ x = 35/14

(c) \dfrac{5(2x - 1) + 3(4x + 3)}{4x - 1} = 5

4x−1

5(2x−1)+3(4x+3)

=5

→ \dfrac{10x - 5 + 12x + 9}{4x - 1} = 5

4x−1

10x−5+12x+9

=5

→ \dfrac{22x + 4}{4x - 1} = 5

4x−1

22x+4

=5

Cross Multiplying, we get

→ 22x + 4 = 5(4x - 1)

→ 22x + 4 = 20x - 5

→ 22x - 20x = - 5 - 4

→ 2x = - 9

→ x = - 9/2

(d) \dfrac{2(3x - 1) - (2x + 1)}{7x - 2} = \dfrac{1}{2}

7x−2

2(3x−1)−(2x+1)

=

2

1

→ \dfrac{6x - 2 - 2x - 1}{7x - 2} = \dfrac{1}{2}

7x−2

6x−2−2x−1

=

2

1

→ \dfrac{4x - 3}{7x- 2} = \dfrac{1}{2}

7x−2

4x−3

=

2

1

Cross Multiplying, we get

→ 2(4x - 3) = 1(7x - 2)

→ 8x - 6 = 7x - 2

→ 8x - 7x = - 2 + 6

→ x = 4

(e) \dfrac{x + 5}{2} + \dfrac{x - 5}{3} = \dfrac{25}{6}

2

x+5

+

3

x−5

=

6

25

→ \dfrac{3(x + 5) + 2(x - 5)}{6} = \dfrac{25}{6}

6

3(x+5)+2(x−5)

=

6

25

→ 3(x + 5) + 2(x - 5) = 25

→ 3x + 15 + 2x - 10 = 25

→ 5x + 5 = 25

→ 5x = 25 - 5

→ 5x = 20

→ x = 20/5

→ x = 4

(f) (5x - 1)(x + 3) = (x - 5)(5x + 1) + 40

→ 5x(x + 3) - 1(x + 3) = x(5x + 1) - 5(5x + 1) + 40

→ 5x² + 15x - x - 3 = 5x² + x - 25x - 5 + 40

→ 15x - x - 3 = x - 25x - 5 + 40

→ 14x - 3 = - 24x + 35

→ 14x + 24x = 35 + 3

→ 38x = 38

→ x = 38/38

→ x = 1

(g) 10 - (\frac{x - 1}{2}) - (\frac{x - 2}{3}) = \frac{x - 3}{4} 10−(

2

x−1

)−(

3

x−2

)=

4

x−3

→ 10 = \dfrac{x - 3}{4} + \dfrac{x - 1}{2} + \dfrac{x - 2}{3} 10=

4

x−3

+

2

x−1

+

3

x−2

→ 10 = \dfrac{3(x - 3) + 6(x - 1) + 4(x - 2)}{12} 10=

12

3(x−3)+6(x−1)+4(x−2)

→ (10)(12) = 3(x - 3) + 6(x - 1) + 4(x - 2)

→ 120 = 3x - 9 + 6x - 6 + 4x - 8

→ 120 = 13x - 23

→ 13x = 120 + 23

→ 13x = 143

→ x = 143/13

→ x = 11

(h) \dfrac{0.5(x - 0.4)}{0.35} - \dfrac{0.6(x - 2.71)}{0.42} =x + 6.1

0.35

0.5(x−0.4)

−

0.42

0.6(x−2.71)

=x+6.1

→ \dfrac{0.5x - 0.20}{0.35} - \dfrac{0.6x - 1.626}{0.42} = x + 6.1

0.35

0.5x−0.20

−

0.42

0.6x−1.626

=x+6.1

→ \dfrac{6(0.5x - 0.20) - 5(0.6x - 1.626)}{2.10} = x + 6.1

2.10

6(0.5x−0.20)−5(0.6x−1.626)

=x+6.1

→ \dfrac{3.0x - 1.20 - 3.0x + 8.13}{2.10} = x + 6.1

2.10

3.0x−1.20−3.0x+8.13

=x+6.1

→ \dfrac{ - 1.20 + 8.13}{2.10} = x + 6.1

2.10

−1.20+8.13

=x+6.1

→ \dfrac{6.93}{2.10} = x + 6.1

2.10

6.93

=x+6.1

→ \dfrac{693}{210} = x + 6.1

210

693

=x+6.1

→ 3.3 = x + 6.1

→ x = 3.3 - 6.1

→ x = - 2.8

(i) (x + 4)² - (x - 5)² = 9

Identity : (a + b)² = a² + b² + 2ab

Here, a = x, b = 4

(a - b)² = a² + b² - 2ab

Here, a = x, b = 5

→ [ (x)² + (4)² + 2(x)(4) ] - [ (x)² + (5)² - 2(x)(5) ] = 9

→ [ x² + 16 + 8x ] - [ x² + 25 - 10x ] = 9

→ x² + 16 + 8x - x² - 25 + 10x = 9

→ 16 + 8x - 25 + 10x = 9

→ - 9 + 18x = 9

→ 18x = 9 + 9

→ 18x = 18

→ x = 18/18

→ x = 1

(j) \dfrac{7x - 1}{4} - \dfrac{1}{7} (2x - \frac{1 - x}{2} ) = 4

4

7x−1

−

7

1

(2x−

2

1−x

)=4

→ \frac{7x - 1}{4} - \dfrac{1}{7} ( \frac{2(2x) - (1 - x)}{2} ) = 4

4

7x−1

−

7

1

(

2

2(2x)−(1−x)

)=4

→ \dfrac{7x - 1}{4} - \dfrac{1}{7} ( \frac{4x - 1 + x}{2} ) = 4

4

7x−1

−

7

1

(

2

4x−1+x

)=4

→ \dfrac{7x - 1}{4} - \dfrac{1}{7} ( \frac{5x - 1}{2} ) = 4

4

7x−1

−

7

1

(

2

5x−1

)=4

→ \dfrac{7x - 1}{4} - \dfrac{5x - 1}{14} = 4

4

7x−1

−

14

5x−1

=4

→ \dfrac{7(7x - 1) - 2(5x - 1)}{28} = 4

28

7(7x−1)−2(5x−1)

=4

→ \dfrac{49x - 7 - 10x + 2}{28} = 4

28

49x−7−10x+2

=4

→ \dfrac{39x - 5}{28} = 4

28

39x−5

=4

→ 39x - 5 = (4)(28)

→ 39x - 5 = 112

→ 39x = 112 + 5

→ 39x = 117

→ x = 117/39

→ x = 3

(k) \dfrac{3x + 5}{4x + 2} = \dfrac{3x + 4}{4x + 7}

4x+2

3x+5

=

4x+7

3x+4

Cross Multiplying, we get

→ (3x + 5)(4x + 7) = (3x + 4)(4x + 2)

→ 3x(4x + 7) + 5(4x + 7) = 3x(4x + 2) + 4(4x + 2)

→ 12x² + 21x + 20x + 35 = 12x² + 6x + 16x + 8

→ 21x + 20x + 35 = 6x + 16x + 8

→ 41x + 35 = 22x + 8

→ 41x - 22x = 8 - 35

→ 19x = - 27

→ x = - 27/19

(l) \dfrac{0.8 + 0.7x}{x} = 0.86

x

0.8+0.7x

=0.86

Cross Multiplying, we get

→ 0.8 + 0.7x = (0.86)(x)

→ 0.8 + 0.7x = 0.86x

→ 0.86x - 0.7x = 0.8

→ 0.16x = 0.8

→ x = 0.8/0.16

→ x = 80/16

→ x = 5