# Exercise 2.2

**Solve the following linear equations.**

**1.**x 2 - 1 5 = x 3 + 1 4

Now, let's solve for x. The denominators are 2,5,3,4. The LCM of these numbers is .

Multiplying every term in the equation by 60: 60 x 2 − 1 5 = 60 x 3 + 1 4 gives us 30 x − 12 = 20 x + 15

Subtracting 20x from both sides: 30 x − 20 x − 12 = 15 which simplifies to - = 15

Adding 12 to both sides of the equation with x : 10 x − 12 + 12 = 15 + 12 which gives us 10 x = 27

Finally dividing both sides by 10: x =

Substituting x in the equation: x 2 − 1 5 = x 3 + 1 4

⇒ 27 10 × 2 - 1 5 = 27 10 × 3 + 1 4

**2.**n 2 - 3 n 4 + 5 n 6 = 21

Simplifying the equation as LCM of 2, 4 and 6 is .

Substituting n in the equation: n 2 − 3 n 4 + 5 n 6

⇒ 36 2 - 3 × 36 4 + 5 × 36 6 ⇒ 18 − 27 + 30 =

**3.**x + 7 - 8 x 3 = 17 6 - 5 x 2

Simplifying the equation: We get LCM of 2, 3 and 6 is .

Multiplying every term in the equation : 6 × x + 7 − 8 x 3 = 6 × 17 6 − 5 x 2

Expanding the equation : 6 x + 42 - 2 x 8 x = 117 − 3 5 x

Simplifying each term : 6 x − 16 x + = − 15 x

Isolating the variable x: + 42 = 17 − 15 x which gives us − 10 x + 15 x + 42 = 17 ⇒ + 42 = 17

Subtracting 42 from both sides: 5x =

Dividing both sides by 5: x = − 25 5 ⇒ x =

Substituting x in equation: − 5 + 7 − 8 − 5 3 = - 5 − 5 2

⇒ − 5 + 7 + 40 3 = 17 6 + 25 2 ⇒ 2 + 40 3 =

Further simplfying: 6 3 + 40 3 = 92 6 ⇒ = 92 6

We get: 92 6 = 46 3 i.e. LHS equal to RHS for x = .

**4.**x − 5 3 = x − 3 5

The LCM of 3 and 5 is .

Multiplying every term in the equation by 15 to simplify: 15 x − 5 3 = 15 x − 3 5 ⇒ x − 5 = x − 3

Expanding: - 25 = 3 x -

Isolating by subtracting 3 x from both sides: 5 x − 3 x − 25 = -9

Adding 25 to both sides: 2 x − 25 + 25 = − 9 + 25 ⇒ 2 x =

Dividing both sides by 2: x = 16 2 which gives us x =

Substituting x in the equation: ( -5)/3 = ( -3)/3 ⇒ /3 = /5 where both further reduce to .

**Thus, LHS is equal to RHS**.

**5.**3 t − 2 4 - 2 t + 3 3 = 2 3 − t

The LCM of 4 and 3 is .

Multiplying every term in the equation by 12 : 12 3 t − 2 4 - 12 2 t + 3 3 = 12 2 3 − 12 t .

Equation simplifies to: 3 3 t − 2 − 4 2 t + 3 = 8 − 12 t

Expanding, we get: 9 t − 6 − 8 t − 12 = 8 − 12 t .

Separating out like terms : t − 18 = 8 − 12 t .

Isolating 't': t + 12 t − 18 = 8 ⇒ - 18 =

Adding 18 to both sides : 13 t − 18 + 18 = 8 + 18 ⇒ 13 t =

Dividing both sides by 13: t = 26 13 ⇒ t =

Substituting 't' in equation: 3 2 − 2 4 − 2 2 + 3 3 = 2 3 − 2

⇒ 3 − 7 3 = 2 − 6 3 ⇒ =

**6.**m − m − 1 2 = 1 − m − 2 3

Simplify the Equation : 2 m − m − 1 2 = 1 − m − 2 3 ⇒ 2 m − m + 1 2 = 1 − m − 2 3 .

⇒ m + 1 2 = 1 − m − 2 3 ⇒ m + 1 2 = 3 − m − 2 3 ⇒ m + 1 2 = 3 − m − 2 3 .

⇒ m + 1 2 = 5 − m 3 ⇒ 3 m + 1 = 2 5 − m ⇒ 3 m + 3 = 2 5 − m ⇒ 3 m + 3 = 10 − 2 m .

⇒ 5 m + 3 = 10 ⇒ 5 m = 10 − 3 ⇒ = ⇒ m = .

Substituting m in equation: 7 5 - 7 5 − 1 2 = 1 - 7 5 − 2 3

⇒ = (LHS = RHS)

**Simplify and solve the following linear equations.**

**7.**3 t − 3 = 5 2 t + 1

Simplifying the equation: 3 t − 9 = 5 2 t + 1 ⇒ 3 t − 9 = 10 t + 5 ⇒ 3 t − 10 t − 9 = 5

Substituting t in equation: 3 − 2 − 3 = 5 2 − 2 + 1

⇒ 3 − 5 = 5 − 3 ⇒ LHS = RHS

**8.**15 y − 4 − 2 y − 9 + 5 y + 6 = 0

Simplifying the equation : 15 − 6 − 2 y + 18 + 5 y + 30 = 0 ⇒ 15 y − 2 y + 5 y − 60 + 18 + 30 = 0 ⇒ = 0

⇒ 18y = 12 ⇒ y =

Substituting y in equation: 15 2 3 − 4 - 2 2 3 − 9 + 5 2 3 + 6 = 0

⇒ 15 × − 10 3 - 2 − 25 3 + 5 20 3 = 0 ⇒ − 150 3 + 50 3 + 100 3 = 0

Thus, − 150 + 50 + 100 3 = i.e. LHS is equal to RHS = .

**9.**3 5 z − 7 − 2 9 z − 11 = 4 8 z − 13 − 17

Simplifying the equation : 15 z − 21 − 18 z + 22 = 4 8 z − 13 − 17 ⇒ − 3 z + 1 = 4 8 z − 13 − 17 ⇒ − 3 z +1 = -

⇒ − 3 z = 32 z − 70 ⇒ = − 70

So, z = − 70 − 35 ⇒ z =

Substituting z in equation: 3 5 × 2 − 7 − 2 9 × 2 − 11 = 4 8 × 2 − 13 − 17

⇒ 3 × 3 − 2 × 7 = 4 × 3 − 17 ⇒ LHS = and RHS =

**10.**0.25 4 f − 3 = 0.05 10 f − 9

Simplifying the equation : 25 100 4 f − 3 = 5 100 10 f − 9 ⇒ 1 4 4 f − 3 = 1 20 10 f − 9 .

⇒ 4 f − 3 4 = 10 f − 9 20 ⇒ 20 4 f − 3 = 4 10 f − 9

⇒ 80 f − 40 f − 60 = − 36 ⇒ f = − 36 + 60 ⇒ f = ⇒ f =

Substituting f in equation: 0.25 4 × 0.6 − 3 = 0.05 10 × 0.6 − 9

⇒ LHS = RHS