Question

1. \\(\frac{-2y - 5}{y + 7} = \frac{5}{2}\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

We have the equation \(\frac{-2y - 5}{y + 7}=\frac{5}{2}\). Cross - multiplying (if \(y
eq - 7\) to avoid division by zero) gives us \(2(-2y - 5)=5(y + 7)\).

Step2: Expand both sides

Expanding the left - hand side: \(2\times(-2y)+2\times(-5)=-4y-10\).
Expanding the right - hand side: \(5\times y+5\times7 = 5y + 35\).
So the equation becomes \(-4y-10 = 5y+35\).

Step3: Move all y - terms to one side and constants to the other

Add \(4y\) to both sides: \(-4y + 4y-10=5y + 4y+35\), which simplifies to \(-10 = 9y+35\).
Then subtract 35 from both sides: \(-10-35=9y+35 - 35\), so \(-45 = 9y\).

Step4: Solve for y

Divide both sides by 9: \(y=\frac{-45}{9}=-5\).
We need to check if \(y = - 5\) makes the original denominator \(y + 7\) non - zero. When \(y=-5\), \(y + 7=-5 + 7 = 2
eq0\), so \(y=-5\) is a valid solution.

Answer:

\(y=-5\)