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Reducing Equations to Linear Form

ax + b 5x + 1 2
Consider an equation of the form = k , say =
cx + d 3 + x 3
This is clearly not a linear equation. However, it can be reduced to a linear equation by using
cross-multiplication and then can be solved easily. In this method we multiply the numerator of
the LHS to the denominator of the RHS. Similarly, multiply the numerator of RHS to the
denominator of LHS and equate both the products.
5x + 1 2 Using cross-multiplication
Now, let us solve = . ax + b k
3 + x 3 Note: =
cx +
1
d
We perform cross-multiplication on the given equation as ⇒ ax + b = k(cx + d)
follows:
5x + 1 2
=
3 + x 3
⇒ 3(5x + 1) = 2(3 + x) ⇒ 15x + 3 = 6 + 2x
3
⇒ 15x – 2x = 6 – 3 ⇒ 13x = 3 ⇒ x = 13

3 15
5x + 1 5 × 13 + 1 13 + 1 15 + 13 13 28 2
Verification: LHS = = = = × = =
3 + x 3 + 3 3 + 3 13 39 + 3 42 3
2 13 13
And RHS =
3
∴ LHS = RHS, hence verified.
2x − 5 − 1
Example 17: Solve the equation: 3x = 4 .
Solution: By cross-multiplication,

2x − 5 −1
= ⇒ 4(2x – 5) = –1(3x)
3x 4
8x – 20 = –3x ⇒ 8x + 3x = 20
20
⇒ 11x = 20 ⇒ x =
11
20 40
2x − 5 2 × 11 − 5 11 − 5 − 15 − 1
Verification: LHS = = = = =
3x 3 × 20 60 60 4
11 11
−1
And RHS = ⇒ LHS = RHS, hence verified.
4
4x − 3 3
Example 18: Solve the equation: = .
2x + 6 2
Solution: By cross-multiplication,
4x − 3 3
=
2x + 6 2
⇒ 2(4x – 3) = 3(2x + 6) ⇒ 8x – 6 = 6x + 18

24
⇒ 8x – 6x = 18 + 6 ⇒ 2x = 24 ⇒ x = = 12
2

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