# Completing the following equation by the suitable method

i got this linear equation two variable problems for my school. I understand the basics of the normal linear equation but this seems different instead having a pure number after the "=" they got a ration, here is the problem.

$$X:2Y = 5:14$$ $$(X+4) : (3Y-21) = 2:3$$

What i've tried to do is just guess the X and Y and to solve it (took a long time and still dont find it) so as example if the X is 5 and the Y is 7, it fit the first equation but wont fit in the 2nd question.

The ":" sign is for ration, not "divided by"

Please answer how you solve and not just give the answer, because as you see im trying to learn not trying to just solve my homework.

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What does $(X+4) : (3Y:21) = 2:3$ mean if "$:$" does not mean "divided by"? – Henry Jul 12 '14 at 10:01
What answer would you get if "$:$" did mean "divided by"? Solve the simultaneous equations and then see if it provides an answer which works with the ratios – Henry Jul 12 '14 at 10:03
I've solve it :) thanks, for the help. – Scott Jul 12 '14 at 10:25
May I encourage you to post your solution as an answer. – Gerry Myerson Jul 12 '14 at 12:32

Ratios are just a shorthand for expressing the relative sizes of quantities to one another. In the case of comparing two quantities, you really aren't gaining anything by using ratio notation. For example $$(X+4):(3Y-21) = 2:3$$ is equivalent to $$\frac{X+4}{3Y-21} = \frac{2}{3}$$

Where ratio notation is appropriate is where there are multiple ratios that need to be stated, for example

• the ratio of blue cars to white cars is 2:3
• the ratio of blue cars to red cars is 1:2
• the ratio of red cars to white cars is 4:3

could be restated more compactly as

the ratio of blue to white to red cars is 2:3:4

Alternatively, we could also have algebraic expressions like $$(X+4):(3Y-21):(2Z-1) = 2:3:5$$ being just a shorthand for the system of equations $$\frac{X+4}{3Y-21} = \frac{2}{3}$$ $$\frac{X+4}{2Z-1} = \frac{2}{5}$$

I hope this explanation was helpful :)

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