Solve the system graphically.
-$x^{2}+y^{2}=7$
$y=\sqrt[3]{x+6}$
A)$\approx(-2,13,157) \text { and } \approx(1,76 \cdot 1.98)$
B)$\approx(-1.76,-1.98) \text { and } \approx(2.13,-1.57)$
C)$\approx(-2.13,-1.57) \text { and } \approx(1.76,-1.98)$
D)$\approx(-2.29,-1.32) \text { and } \approx(2.29,1.32)$

Question

Quizplus · Accepted Answer

To solve the system of equations graphically, we need to plot both equations on the same set of axes and find their points of intersection. The equations given are:
1. $x^{2}+y^{2}=7$
2. $y=\sqrt[3]{x+6}$
The first equation represents a circle with a radius of $\sqrt{7}$ centered at the origin (0,0). The second equation represents a cubic root function.
To find the approximate points of intersection, we would typically use graphing software or a graphing calculator. However, since we are looking for an analytical solution, we can analyze the given options to see which one fits the system of equations.
Let's evaluate the options provided:
A) $\approx(-2.13, 1.57) \text{ and } \approx(1.76 \cdot 1.98)$
B) $\approx(-1.76,-1.98) \text{ and } \approx(2.13,-1.57)$
C) $\approx(-2.13,-1.57) \text{ and } \approx(1.76,-1.98)$
D) $\approx(-2.29,-1.32) \text{ and } \approx(2.29,1.32)$
We can quickly check the validity of these points by substituting them into both equations. The points must satisfy both equations to be considered solutions.
Let's check option A:
For the point $(-2.13, 1.57)$:
1. $x^{2}+y^{2} = (-2.13)^{2} + (1.57)^{2} \approx 4.5369 + 2.4649 = 7.0018$, which is close to 7.
2. $y = \sqrt[3]{x+6} = \sqrt[3]{-2.13+6} \approx \sqrt[3]{3.87} \approx 1.57$, which matches the y-value.
For the point $(1.76, 1.98)$:
1. $x^{2}+y^{2} = (1.76)^{2} + (1.98)^{2} \approx 3.0976 + 3.9204 = 7.018$, which is close to 7.
2. $y = \sqrt[3]{x+6} = \sqrt[3]{1.76+6} \approx \sqrt[3]{7.76} \approx 1.98$, which matches the y-value.
However, there seems to be a typo in option A. The point should be $(1.76, 1.98)$ instead of $(1.76 \cdot 1.98)$. Assuming this is a typographical error, option A provides points that are close to satisfying both equations, which suggests that these points could be the correct graphical solution.
Without graphing tools, we cannot confirm the exact points of intersection, but based on the given options and the quick check above, option A seems to be the most plausible solution, with the correction to the second point being $(1.76, 1.98)$.

Solve the System Graphically x2+y2=7x^{2}+y^{2}=7x2+y2=7 y=x+63y=\sqrt[3]{x+6}y=3x+6​ A) ≈(−2,13,157) and ≈(1,76⋅1.98)\approx(-2,13,157) \text { and } \approx(1,76 \cdot 1.98)≈(−2,13,157) and ≈(1,76⋅1.98)

Solve the System Graphically $x^{2}+y^{2}=7$ $y=\sqrt[3]{x+6}$ A) $\approx(-2,13,157) \text { and } \approx(1,76 \cdot 1.98)$