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]
