How to solve a literal equation How do I

Oliver Tyler

Oliver Tyler

Answered question

2022-04-01

How to solve a literal equation
How do I solve 2x1=3x+a I cannot solve it and have spent an hour on it trying many different ways. Please help me! Thank you!

Answer & Explanation

Lesly Fernandez

Lesly Fernandez

Beginner2022-04-02Added 16 answers

To get the variables in a manageable place, take the logarithm of both sides of the equation (it does not matter what base you use; the power law for logarithms will let you bring the powers in front of the logarithm):
2x1=3x+aln2x1=ln3x+a(x1)ln2=(x+a)ln3
The above is valid, since logarithm functions are one-to-one, and since the first equation above has positive quantities on both sides.
Generally, if you have an equation with the variable appearing in an exponent, you can try (perhaps after a bit of algebra) taking logarithms to produce a more manageable equation as in the case above.
Finishing this problem:
(x1)ln2=(x+a)ln3
xln2ln2=xln3+aln3
xln2xln3=ln2+aln3
x(ln2ln3)=ln2+aln3
x=ln2+aln3ln2-ln3
=ln(2·3a)ln(2/3)

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