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Equations Reducible to the Linear Form

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Example

Solve: `(x + 1)/(2x + 3) = 3/8`.

`(x + 1)/(2x + 3) = 3/8`.

We multiply both sides of the equation by (2x + 3),

`((x + 1)/(2x + 3)) xx (2x + 3) = 3/8 xx (2x + 3)`

(2x + 3) gets cancelled on the LHS we have then,

`x + 1 = (3(2x + 3))/8`

Multiplying both sides by 8,
8(x + 1) = 3(2x + 3)
or 8x + 8 = 6x + 9
or 8x = 6x + 9 - 8
or 8x = 6x + 1
or 8x - 6x = 1
or 2x = 1
or x = `1/2`
The solution is x = `1/2`.
Check:
Number of LHS = `1/2 + 1 = (1 + 2)/2 = 3/2`
Denominator of LHS = 2x + 3 = 2 × `1/2` + 3 = 1 + 3 = 4.
LHS = numerator ÷ denominator = `3/2 ÷ 4 = 3/2 xx 1/4 = 3/8.`
LHS = RHS.

Example

Present ages of Anu and Raj are in the ratio 4: 5. Eight years from now the ratio of their ages will be 5: 6. Find their present ages.

Let the present ages of Anu and Raj be 4x years and 5x years respectively.

After eight years, Anu's age = (4x + 8) years;
After eight years, Raj's age = (5x + 8) years.

Therefore, the ratio of their ages after eight years = `(4x + 8)/(5x + 8)`

This is given to be 5: 6

Therefore,

`(4x + 8)/(5x + 8) = 5/6`

Cross-multiplication gives

6(4x + 8) = 5(5x + 8)

or 24x + 48 = 25x + 40

or 24x + 48 - 40 = 25x

or 24x + 8 = 25x

or 8 = 25x - 24x

or 8 = x

Therefore,
Anu's present age = 4x = 4 × 8 = 32 years.
Raj's present age = 5x = 5 × 8 = 40 years.

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