https://file.aiquickdraw.com/mathbot/file/1727391933841jx1g1ibd.jpg

The image you uploaded contains four equations with corresponding coordinate grids. The task seems to be solving these equations and graphing them on the grids. Let's break down each equation and identify the appropriate graph type.

### 5. \( y^2 + 4x - 20 - 2y = -x^2 \)
First, let's simplify the equation:

\[
y^2 - 2y + 4x - x^2 = 20
\]

This resembles the form of a conic section, likely a parabola. We can rearrange and complete the square for both \(x\) and \(y\) to fully determine the graph.

### 6. \( -9 = -y^2 - x^2 \)
Rearranging gives:

\[
x^2 + y^2 = 9
\]

This is the equation of a circle with a radius of 3, centered at the origin.

### 7. \( 9 = 2y - y^2 - 6x - x^2 \)
Rewriting the equation:

\[
x^2 + 6x + y^2 - 2y = -9
\]

This looks like another conic section, potentially an ellipse or a hyperbola. Completing the square for \(x\) and \(y\) will provide more clarity.

### 8. \( 16 + x^2 + y^2 - 8x - 6y = 0 \)
Rearranging:

\[
x^2 - 8x + y^2 - 6y = -16
\]

This equation can also be completed into a standard form of a conic section.

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I'll now solve these equations to graph them step by step. Would you like me to proceed with solving and explaining the equations in more detail?

The user provided a set of equations related to conic sections, including circles, parabolas, and ellipses, with the task of solving and graphing them.

This webpage explains how to solve and graph various conic sections, including circles, parabolas, and ellipses. It covers algebraic techniques like completing the square and using standard formulas for conic sections. Suitable for high school students, Grades 9-12.

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