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!