Final answer:
To find the intersections, substitute y from the line equation y = x - 3 into the circle equation x^2 + y^2 = 29 and solve for x. Then, use the found x-values to solve for the corresponding y-values. The points of intersection are (-2, -5) and (5, 2).
Step-by-step explanation:
To find the intersection of the circle x^2 + y^2 = 29 and the line y = x - 3, we substitute the expression for y from the line equation into the circle equation.
Substitute y = x - 3 into x^2 + y^2 = 29:
x^2 + (x - 3)^2 = 29.
Expand the squared term and simplify:
x^2 + x^2 - 6x + 9 = 29.
Combine like terms and solve for x:
2x^2 - 6x - 20 = 0.
Factor the quadratic equation or use the quadratic formula to find the x-values.
(Let's assume it factored to (2x+4)(x-5) = 0, then x can be -2 or 5).
Substitute the x-values into y = x - 3 to find corresponding y-values:
For x = -2, y = -5 and
for x = 5, y = 2.
The points of intersection are (-2, -5) and (5, 2).