To solve for [tex]\( y \)[/tex] in the equation:
[tex]\[ y - 11 = -9 + \frac{24}{y} \][/tex]
we can follow these steps:
1. Isolate the variable terms: Move the terms involving [tex]\( y \)[/tex] on one side of the equation.
Start by adding 11 to both sides to simplify the left side:
[tex]\[ y - 11 + 11 = -9 + \frac{24}{y} + 11 \][/tex]
Simplifying the right side:
[tex]\[ y = 2 + \frac{24}{y} \][/tex]
2. Eliminate the fraction: Multiply every term by [tex]\( y \)[/tex] to remove the fraction.
[tex]\[ y \cdot y = y \cdot \left( 2 + \frac{24}{y} \right) \][/tex]
Distribute the [tex]\( y \)[/tex]:
[tex]\[ y^2 = 2y + 24 \][/tex]
3. Rewrite in standard quadratic form:
[tex]\[ y^2 - 2y - 24 = 0 \][/tex]
4. Factor the quadratic equation:
We need two numbers that multiply to [tex]\(-24\)[/tex] and add up to [tex]\(-2\)[/tex]. Those numbers are 4 and -6.
[tex]\[ y^2 - 2y - 24 = (y - 6)(y + 4) = 0 \][/tex]
5. Find the solutions: Set each factor equal to zero.
[tex]\[ y - 6 = 0 \quad \text{or} \quad y + 4 = 0 \][/tex]
6. Solve for each factor:
[tex]\[ y = 6 \quad \text{or} \quad y = -4 \][/tex]
Thus, the solutions to the equation are:
[tex]\[ y = -4, \, 6 \][/tex]
