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15. If [tex]$x + 2y = a$[/tex] and [tex]$x - 2y = b$[/tex], which of the following is an expression for [tex][tex]$\frac{x}{y}$[/tex][/tex]?

A) [tex]$2\left(\frac{a+b}{a-b}\right)$[/tex]

B) [tex]$2\left(\frac{a-b}{a+b}\right)$[/tex]

C) 1

D) None of these

Sagot :

GinnyAnsweridn
Given the equations:

1. [tex]\( x + 2y = a \)[/tex]
2. [tex]\( x - 2y = b \)[/tex]

We need to find an expression for [tex]\(\frac{x}{y}\)[/tex].

First, we add the two equations to eliminate [tex]\( y \)[/tex]:

[tex]\[ (x + 2y) + (x - 2y) = a + b \][/tex]
[tex]\[ x + x = a + b \][/tex]
[tex]\[ 2x = a + b \][/tex]
[tex]\[ x = \frac{a + b}{2} \][/tex]

Next, we subtract the second equation from the first to eliminate [tex]\( x \)[/tex]:

[tex]\[ (x + 2y) - (x - 2y) = a - b \][/tex]
[tex]\[ x + 2y - x + 2y = a - b \][/tex]
[tex]\[ 4y = a - b \][/tex]
[tex]\[ y = \frac{a - b}{4} \][/tex]

Now, we find the ratio [tex]\(\frac{x}{y}\)[/tex]:

[tex]\[ \frac{x}{y} = \frac{\left(\frac{a + b}{2}\right)}{\left(\frac{a - b}{4}\right)} \][/tex]

To simplify, we divide by a fraction by multiplying by its reciprocal:

[tex]\[ \frac{x}{y} = \left(\frac{a + b}{2}\right) \times \left(\frac{4}{a - b}\right) \][/tex]
[tex]\[ \frac{x}{y} = \frac{4(a + b)}{2(a - b)} \][/tex]
[tex]\[ \frac{x}{y} = \frac{2(a + b)}{a - b} \][/tex]

The correct expression for [tex]\(\frac{x}{y}\)[/tex] is:

A) [tex]\( 2\left(\frac{a+b}{a-b}\right) \)[/tex]
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