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.
 A: 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 :)
A: We can start by solving both equations for the same variable.
$$X:2Y = 5:14\implies Y = \frac{7 X}{5}$$
$$(X+4) : (3Y-21) = 2:3 \implies Y = \frac{X + 18}{2}$$
We now equate the two "solutions" of $Y$.
$$\frac{7 X}{5}=\frac{X + 18}{2}\\
\implies \frac{7 X}{5} = \frac{X}{2} + 9\\\implies 14X-5X  =90\\
\implies X=10\qquad
\implies  Y=\frac{7 X}{5}
=\frac{70}{5}=14$$
