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I have this given problem. That asking me to solve for $x$. Although this example has answered. I've had troubles on a certain part.

Here's the equation with answerenter image description here

\begin{align*} \frac{2x-a}b &= \frac{4x-b}a\\ a(2x-a) &= b(4x-b) \\ 2ax-a^2 &= 4bx - b^2 \\ 2ax-4bx &= a^2 - b^2 \\ x(2a-4b) &= a^2 - b^2 \\ x&=\frac{a^2-b^2}{2a-4b} \end{align*}

How did we arrive to

$$a(2x -a) = b(4x-b)?$$

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    Multiply the first equation by $ab$. – 2017-01-15
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    Cross-multiplying – 2017-01-15
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    @KyelJmD You see the check mark at the left side of our answers? You can click it to which answer you want to accept. Thank you. – 2017-01-15

5 Answers 5

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The process done is called Cross-multiplication.

This technique is generally used to work faster, we can solve such problems in 1 step rather than 2.

Let's say we have the following equation:

$$\frac{a}{b}=\frac{c}{d}$$

We can do it this way, without cross-multiplication:

Multiply both sides by $b$: $$a=\frac{c}{d} b$$ Multiply both sides by $d$: $$ad=bc$$

Now, let's use cross-multiplication:

$$\frac{\color{red}a}{\color{blue}b}=\frac{\color{blue}c}{\color{red}d}$$

We multiply the red and blue terms only.

$$\color{red}{ad}=\color{blue}{bc}$$

Now apply the same technique to the equation in your picture, and you should get the answer you were looking for.

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    How would I know if I should cross multiplication in such situations? or what is an indicator that I should use cross multiplication? – 2017-01-21
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    @KyeIJmD Generally when you have fractions on both sides it is a method which should definetely come into consideration. – 2017-01-21
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Maybe we need to write again the equation:

$$\frac{2x-a}{b}=\frac{4x-b}{a}.$$

Using the Multiplication Property of Equality, we get

$$ab\cdot\left(\frac{2x-a}{b}\right)=ab\cdot\left(\frac{4x-b}{a} \right).$$ Simplifying, we get $$\frac{a\cdot b\cdot (2x-a)}{b}=\frac{a\cdot b\cdot (4x-b)}{a}.$$ Apply Cancellation Law in Multiplication (meaning we can cancel $b$ at the left hand side and same to $a$ at the right hand side), we get $$a\cdot(2x-a)=b\cdot(4x-b).$$ Hope this help.

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Just cross multiply terms.

(2x - a) * a = (4x - b) * b

Which can be written as -

a(2x - a) = b(4x - b)

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    If any doubt let me know. – 2017-01-15
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Multiply by $ab$ on both sides.

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    can you show the step by step process? – 2017-01-15
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    @KyelJmD $ab(\dfrac{2x-a}{b})=ab(\dfrac{4x-b}{a})$ – 2017-01-15
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    @KyelJmD $a{\color{red}b}\dfrac{2x-a}{{\color{red}b}}={\color{red}a}b\dfrac{4x-b}{{\color{red}a}}$ – 2017-01-15
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    @KyelJmD delete red characters – 2017-01-15
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    @KyelJmD ${\color{blue}a}(2x-a)={\color{blue}b}(4x-b)$ – 2017-01-15
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    @MyGlasses last question,you're deleting/cancelling the red characters, does that mean you're dividing them? or is it a short cut method for something else? – 2017-01-15
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    @KyelJmD Indeed we divided them i.e $\dfrac{\color{red}b}{\color{red}b}=1$ and we don't write $1$ instead of them after deletion. – 2017-01-15
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As other have called it: cross-multiplying (or taking cross-products).

The thing I wanted to add to this answer is that, to my significant embarrassment, I just realized the other day that this is in fact the formal definition of fraction equality: $\frac{a}{b} = \frac{c}{d} \Leftrightarrow ad = bc$. (Which makes sense: operations on fractions need to be defined in terms of some pre-existing entity, namely integers.)

So another way of putting it is: we are directly applying the definition of equality for fractions.

https://en.wikipedia.org/wiki/Rational_number#Formal_construction