To solve the equation
[tex]\[
\frac{8}{y - 2} = \frac{4}{4y + 6}
\][/tex]
we need to find the value of [tex]\( y \)[/tex] that satisfies it. Follow these steps to solve it:
1. Cross-Multiply: When we have an equation of the form [tex]\( \frac{a}{b} = \frac{c}{d} \)[/tex], we can cross-multiply to get
[tex]\[
a \cdot d = b \cdot c
\][/tex]
Applying this to our equation:
[tex]\[
8 \cdot (4y + 6) = 4 \cdot (y - 2)
\][/tex]
2. Distribute: Distribute the constants through the parentheses:
[tex]\[
8 \cdot 4y + 8 \cdot 6 = 4 \cdot y - 4 \cdot 2
\][/tex]
[tex]\[
32y + 48 = 4y - 8
\][/tex]
3. Isolate [tex]\( y \)[/tex]: Let's collect all terms involving [tex]\( y \)[/tex] on one side of the equation and constant terms on the other side:
[tex]\[
32y + 48 = 4y - 8
\][/tex]
Subtract [tex]\( 4y \)[/tex] from both sides:
[tex]\[
32y - 4y + 48 = -8
\][/tex]
[tex]\[
28y + 48 = -8
\][/tex]
Subtract 48 from both sides:
[tex]\[
28y = -8 - 48
\][/tex]
[tex]\[
28y = -56
\][/tex]
4. Solve for [tex]\( y \)[/tex]: Divide both sides by 28:
[tex]\[
y = \frac{-56}{28}
\][/tex]
[tex]\[
y = -2
\][/tex]
Therefore, the solution to the equation is
[tex]\(
y = -2
\)[/tex]
Hence,
[tex]\[
y = -2
\][/tex]
