To evaluate the expression [tex]\(\frac{x^2 - y}{y^2 - x}\)[/tex] for [tex]\(x = 5\)[/tex] and [tex]\(y = 10\)[/tex], we need to follow a step-by-step process:
1. Substitute the values into the expression:
Given [tex]\(x = 5\)[/tex] and [tex]\(y = 10\)[/tex], we substitute these values into the expression:
[tex]\[
\frac{x^2 - y}{y^2 - x} \implies \frac{5^2 - 10}{10^2 - 5}
\][/tex]
2. Calculate [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex]:
[tex]\[
x^2 = 5^2 = 25
\][/tex]
[tex]\[
y^2 = 10^2 = 100
\][/tex]
3. Evaluate the numerator, [tex]\(x^2 - y\)[/tex]:
[tex]\[
x^2 - y = 25 - 10 = 15
\][/tex]
4. Evaluate the denominator, [tex]\(y^2 - x\)[/tex]:
[tex]\[
y^2 - x = 100 - 5 = 95
\][/tex]
5. Form the fraction with the evaluated numerator and denominator:
[tex]\[
\frac{x^2 - y}{y^2 - x} = \frac{15}{95}
\][/tex]
6. Simplify the fraction (if possible):
The fraction [tex]\(\frac{15}{95}\)[/tex] simplifies to [tex]\(\frac{3}{19}\)[/tex] when divided by the greatest common divisor, which is 5.
7. Evaluate the simplified fraction:
[tex]\[
\frac{3}{19} \approx 0.15789473684210525
\][/tex]
Therefore, the evaluated expression [tex]\(\frac{x^2 - y}{y^2 - x}\)[/tex] for [tex]\(x = 5\)[/tex] and [tex]\(y = 10\)[/tex] is approximately [tex]\(0.15789473684210525\)[/tex].
