To solve for the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given by the expressions [tex]\( x = 5 - 2 \sqrt{7} \)[/tex] and [tex]\( y = -9 + 2 \sqrt{15} \)[/tex], follow these steps:
1. Expression for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - 2 \sqrt{7}
\][/tex]
We need to evaluate this expression numerically.
- First, we find the value of [tex]\( \sqrt{7} \)[/tex]. We know:
[tex]\[
\sqrt{7} \approx 2.6457513110645906
\][/tex]
- Next, calculate [tex]\( 2 \sqrt{7} \)[/tex]:
[tex]\[
2 \sqrt{7} = 2 \times 2.6457513110645906 \approx 5.291502622129181
\][/tex]
- Now, substitute this value back into the expression for [tex]\( x \)[/tex]:
[tex]\[
x = 5 - 5.291502622129181 \approx -0.29150262212918143
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is approximately [tex]\( -0.29150262212918143 \)[/tex].
2. Expression for [tex]\( y \)[/tex]:
[tex]\[
y = -9 + 2 \sqrt{15}
\][/tex]
We need to evaluate this expression numerically.
- First, we find the value of [tex]\( \sqrt{15} \)[/tex]. We know:
[tex]\[
\sqrt{15} \approx 3.872983346207417
\][/tex]
- Next, calculate [tex]\( 2 \sqrt{15} \)[/tex]:
[tex]\[
2 \sqrt{15} = 2 \times 3.872983346207417 \approx 7.745966692414834
\][/tex]
- Now, substitute this value back into the expression for [tex]\( y \)[/tex]:
[tex]\[
y = -9 + 7.745966692414834 \approx -1.254033307585166
\][/tex]
Thus, the value of [tex]\( y \)[/tex] is approximately [tex]\( -1.254033307585166 \)[/tex].
In summary, the numerical solutions for the given expressions are:
[tex]\[
x \approx -0.29150262212918143
\][/tex]
[tex]\[
y \approx -1.254033307585166
\][/tex]
