To find the approximate solution for the given system of equations:
[tex]\[
\begin{array}{l}
y = x^2 + 5x + 3 \\
y = \sqrt{2x + 5}
\end{array}
\][/tex]
we want to find a pair [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
Step-by-step solution:
1. Write down the system of equations:
[tex]\[
y = x^2 + 5x + 3
\][/tex]
[tex]\[
y = \sqrt{2x + 5}
\][/tex]
2. Consider substituting the second equation into the first:
Substitute [tex]\( y = \sqrt{2x + 5} \)[/tex] into [tex]\( y = x^2 + 5x + 3 \)[/tex]:
[tex]\[
\sqrt{2x + 5} = x^2 + 5x + 3
\][/tex]
3. Solve the resulting equation:
This equation is complex to solve algebraically due to the square root and quadratic terms.
4. Analyze the possible solutions:
We are given choices, and we need to determine which pair approximately satisfies both original equations:
- Option A: (2.1, 0.2)
- [tex]\(y = 2.1^2 + 5(2.1) + 3 = 4.41 + 10.5 + 3 = 17.91\)[/tex]
- [tex]\(y = \sqrt{2(2.1) + 5} = \sqrt{4.2 + 5} = \sqrt{9.2} \approx 3.0\)[/tex]
- This does not satisfy either equation well.
- Option B: (-0.2, 2.1)
- [tex]\(y = (-0.2)^2 + 5(-0.2) + 3 = 0.04 - 1 + 3 = 2.04\)[/tex]
- [tex]\(y = \sqrt{2(-0.2) + 5} = \sqrt{-0.4 + 5} = \sqrt{4.6} \approx 2.14\)[/tex]
- This approximately satisfies both equations well.
- Option C: (-2.1, 0.2)
- [tex]\(y = (-2.1)^2 + 5(-2.1) + 3 = 4.41 - 10.5 + 3 = -3.09\)[/tex]
- [tex]\(y = \sqrt{2(-2.1) + 5} = \sqrt{-4.2 + 5} = \sqrt{0.8} \approx 0.89\)[/tex]
- This does not satisfy either equation well.
- Option D: (0.2, 2.1)
- [tex]\(y = (0.2)^2 + 5(0.2) + 3 = 0.04 + 1 + 3 = 4.04\)[/tex]
- [tex]\(y = \sqrt{2(0.2) + 5} = \sqrt{0.4 + 5} = \sqrt{5.4} \approx 2.32\)[/tex]
- This does not satisfy either equation well.
After analyzing the given options, we see that the pair [tex]\((-0.2, 2.1)\)[/tex] from Option B most closely approximates the solution to the system of equations.
Therefore, the correct answer is:
[tex]\[
\boxed{(-0.2, 2.1)}
\][/tex]
