Cross multiplying vs the long way (doing the same thing to each side) to solve for x

math_knight

New member
from OPen Stax Precalculus section 4.6 # 45

This is a log problem but the mistake I am making is basic algebra...solving for x when x is in the denominator

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The problem is

ln(3) - ln (3-3x)= ln (4)

quotient law of logs leads to

ln(3/3-3x) = ln (4)

3/3-3x = 4

so far so good....

######################
A.) the long way..where did I go wrong?

3/3-3x = 4

divide both sides by 3

3-3x = 4/3

subtract 3 from both sides

-3x = 4/3 - 9/3

-3x = -5/3

x = 5/9

B.) the cross multiply way (I got this correct)

ln(3/3-3x) = ln (4)

3/3-x = 4

12-12x=3

-12x = -9

x= 3/4 CORRECT

the answer in the book is x=3/4

MarkFL

Super Moderator
Staff member
In your long way, when you divided by 3 in your first step, you inverted the left side after the division. You should have:

$$\displaystyle \frac{1}{3-3x}=\frac{4}{3}$$

My first step would have been to reduce the LHS:

$$\displaystyle \frac{1}{1-x}=4$$

Then multiply though by $$\displaystyle \frac{1-x}{4}$$ to get:

$$\displaystyle \frac{1}{4}=1-x$$

$$\displaystyle x=\frac{3}{4}$$

Denis

Senior Member
quotient law of logs leads to

ln(3/3-3x) = ln (4)
No. Left side should be ln(3/(3-3x))

Dr.Peterson

Elite Member
from OPen Stax Precalculus section 4.6 # 45

This is a log problem but the mistake I am making is basic algebra...solving for x when x is in the denominator

########################
The problem is

ln(3) - ln (3-3x)= ln (4)

quotient law of logs leads to

ln(3/(3-3x)) = ln (4)

3/(3-3x) = 4

so far so good....

######################
A.) the long way..where did I go wrong?

3/(3-3x) = 4

divide both sides by 3

3-3x = 4/3

subtract 3 from both sides

-3x = 4/3 - 9/3

-3x = -5/3

x = 5/9

B.) the cross multiply way (I got this correct)

ln(3/(3-3x)) = ln (4)

3/(3-x) = 4

12-12x=3

-12x = -9

x= 3/4 CORRECT

the answer in the book is x=3/4
Note that the parentheses I added are necessary in order to communicate what you mean when you write fractions inline.

A good alternative to cross-multiplication (in fact, the reason cross-multiplication works) is to eliminate fractions by multiplying by the LCD. Your long way actually introduced a fraction that wasn't there already, and played with fractions in a way that led to a mistake.

I would do this (showing more steps than I would actually write):

$$\displaystyle \displaystyle \frac{3}{3-3x} = 4$$

$$\displaystyle \displaystyle \frac{3}{3(1-x)} = 4$$

$$\displaystyle \displaystyle \frac{1}{1-x} = 4$$

$$\displaystyle \displaystyle \frac{1}{1-x}\cdot (1-x) = 4(1-x)$$

$$\displaystyle \displaystyle 1 = 4-4x$$

$$\displaystyle \displaystyle 1+4x = 4$$

$$\displaystyle \displaystyle 4x = 3$$

$$\displaystyle \displaystyle x = \frac{3}{4}$$

This approach allows you to avoid fractions until the end, which is wise for many students!

JeffM

Elite Member
First, as denis said, you made a mistake with parentheses.

$$\displaystyle ln(3) - ln(3 - 3x) = ln(4) \implies ln(3\ / \ \{3 - 3x\} )= ln(4) \implies 3 \ / \ (3 - 3x) = 4.$$

When you are typing, you may have to use in-line conventions, but, working with pencil and paper, you can avoid such mistakes this way

$$\displaystyle ln(3) - ln(3 - 3x) = ln(4) \implies \ln \left ( \dfrac{3}{3 - 3x} \right ) = ln(4) \implies \dfrac{3}{3- 3x} = 4.$$

Working in line requires very careful attention to parentheses, and I avoid in-line when possible (such as when using paper and pencil). When you must work in-line, you can avoid errors when nesting expressions by using [ ] and { } in addition to ( ).

Like Dr. Peterson, I recognize that students have trouble with fractions, particularly algebraic fractions. Therefore, I recommend that you clear fractions as soon as possible by multiplying both sides of any equation with fractions by the least common multiple of the denominators.

$$\displaystyle \dfrac{3}{3- 3x} = 4 \implies (3 - 3x) * \dfrac{3}{3- 3x} = 4(3 - 3x) \implies \\ 3 = 12 - 12x \implies 12x = 12 - 3 \implies x = \dfrac{9}{12} = \dfrac{3}{4}.$$

Be careful with nested expressions; clear fractions early.

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Denis

Senior Member
I insist on cross-multiplication!

3 / (3 - 3x) = 4
4(3 - 3x) = 3
12 - 12x = 3
12x = 9
x = 9/12 = 3/4

math_night: if a/b = c/d then ad = bc.
Usually saves time on a timed test...

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Dr.Peterson

Elite Member
I insist on cross-multiplication!

3 / (3 - 3x) = 4
4(3 - 3x) = 3
12 - 12x = 3
12x = 9
x = 9/12 = 3/4

math_night: if a/b = c/d then ad = ac.
Usually saves time on a timed test...
Usually, yes. But it's a good idea to consider other possibilities when the denominators are more complicated, because cross-multiplication amounts to using the product of the denominators as the common denominator, which can result in having a higher degree than necessary in subsequent steps (e.g. having to solve a cubic). For that reason, it shouldn't be an automatic reflex, as I find it is for many students (because it's so memorable).

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Denis

Senior Member
But you meant ad = bc, right? Quick and easy sometimes means you turn off your brain ... but math-knight did fine with that method.
RIGHT!(edited...thanks DrP)
...was still on my 1st coffee

math_knight

New member
Thank you all, good points and tips...I gotta find the karma button on this site if there is one. Thanks again, I see my mistake in the math problem.