https://file.aiquickdraw.com/mathbot/file/1727541161450idy3fxa3.jpg

The problem in the image asks to find the solution set of a system of equations by graphing them:

1. \((x - 7)^2 + (y + 2)^2 = 49\) — This is the equation of a circle.
2. \(y = x - 2\) — This is the equation of a line.

### Step 1: Analyze the Circle Equation
\[
(x - 7)^2 + (y + 2)^2 = 49
\]
- This is a circle centered at \((7, -2)\) with a radius of \(\sqrt{49} = 7\).

### Step 2: Analyze the Line Equation
\[
y = x - 2
\]
- This is a straight line with a slope of 1 and a y-intercept of -2.

### Step 3: Find the Points of Intersection
To find where the line intersects the circle, substitute \(y = x - 2\) into the circle’s equation:

\[
(x - 7)^2 + (x - 2 + 2)^2 = 49
\]

This simplifies to:

\[
(x - 7)^2 + (x)^2 = 49
\]

Expand both terms:

\[
(x^2 - 14x + 49) + x^2 = 49
\]

Combine like terms:

\[
2x^2 - 14x + 49 = 49
\]

Simplify:

\[
2x^2 - 14x = 0
\]

Factor out \(2x\):

\[
2x(x - 7) = 0
\]

Thus, \(x = 0\) or \(x = 7\).

### Step 4: Find Corresponding y-values
- For \(x = 0\): \(y = 0 - 2 = -2\).
- For \(x = 7\): \(y = 7 - 2 = 5\).

### Solution Set
The points of intersection are \((0, -2)\) and \((7, 5)\).

Would you like more details or have any further questions?

Here are 5 related questions for further understanding:
1. How do you graph a circle based on its equation?
2. What is the geometric meaning of the radius and center of a circle?
3. How do you find the slope of a line from its equation?
4. What are the algebraic methods to solve systems involving a circle and a line?
5. How do you check if a point lies on a circle or line by substituting values?

**Tip:** For solving systems involving circles and lines, substitution is often a powerful method to reduce the system to one variable!

(x - 7)^2 + (y + 2)^2 = 49 and y = x - 2. Find the solution set by graphing both equations and finding all points of intersection.

Learn how to solve a system of equations involving a circle and a line. The problem requires graphing both equations and finding their points of intersection, with detailed steps for algebraic and geometric methods. Suitable for students in Grades 9-12.

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