How do you use cross products to solve 3/4=x/(x+3)?

3 Answers
Aug 5, 2016

x=9

Explanation:

3/4=x/(x+3)
or
4x=3x+9
or
4x-3x=9
or
x=9

Aug 5, 2016

x = 9

Explanation:

To use cross products or color(blue)"cross multiplication" as it is also named.

color(red)(3)/color(blue)(4)=color(blue)(x)/color(red)(x+3)

now multiply the terms in color(blue)("blue")" and "color(red)("red") (X) and equate them.

rArrcolor(blue)(4x)=color(red)(3(x+3))

distribute the bracket : 4x = 3x + 9

subtract 3x from both sides to solve for x

4x-3x=cancel(3x)+9cancel(-3x)rArrx=9

Aug 5, 2016

This is why the cross product works!!!

Explanation:

The cross product is a shortcut that bypasses some stages in solving by first principles. I will use first principles so you can see where the shortcut takes over.

A fraction is split up into two parts. Using descriptive but ul("unconventional names") we have ("count")/("size indicator")

When you wish to ul(directly") compare quantities the "size indicators" have to be the same. This is also true for fractional addition and subtraction. You can not ul("directly") apply addition or subtraction unless the "size indicators" are the same.

" "("count")/("size indicator") ->("numerator")/("denominator")
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Solving your question")

Using a 'common denominator' of 4(x+3)

color(brown)([3/4xx1]=[x/(x+3)xx1]color(blue)(""->""[3/4xx(x+3)/(x+3) ]=[x/(x+3)xx4/4]

(3(x+3))/(4(x+3)) = (4x)/(4(x+3))

If you look at the numerators you will see the result you get by the short cut

Multiply both sides by 4(x+3) and you end up with

3(x+3)=4x" "larr" the consequences of the shortcut"