Graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. (x − 2)²+(y+3)² = 4, y = x - 3

Verified step by step guidance

1. Identify the two equations given: the first is a circle equation \(\left(x - 2\right)^2 + \left(y + 3\right)^2 = 4\), and the second is a line equation \(y = x - 3\).
2. Substitute the expression for \(y\) from the line equation into the circle equation. Replace \(y\) with \(x - 3\) in \(\left(x - 2\right)^2 + \left(y + 3\right)^2 = 4\) to get \(\left(x - 2\right)^2 + \left((x - 3) + 3\right)^2 = 4\).
3. Simplify the equation after substitution: note that \(((x - 3) + 3)\) simplifies to \(x\), so the equation becomes \(\left(x - 2\right)^2 + x^2 = 4\).
4. Expand the squared terms and combine like terms to form a quadratic equation in terms of \(x\). That is, expand \(\left(x - 2\right)^2\) to \(x^2 - 4x + 4\) and add \(x^2\), resulting in \(x^2 - 4x + 4 + x^2 = 4\).
5. Solve the quadratic equation for \(x\), then substitute each solution back into \(y = x - 3\) to find the corresponding \(y\) values. These \((x, y)\) pairs are the points of intersection. Finally, verify each point by plugging them into both original equations to confirm they satisfy both.

Key Concepts

Graphing Circles

A circle's equation in the form (x - h)² + (y - k)² = r² represents a circle centered at (h, k) with radius r. Graphing involves plotting the center and using the radius to mark points equidistant from the center, forming the circle's curve.

Graphing Linear Equations

A linear equation like y = mx + b graphs as a straight line with slope m and y-intercept b. Plotting the intercept and using the slope to find additional points helps visualize the line on the coordinate plane.

Finding Points of Intersection

Points of intersection between two graphs satisfy both equations simultaneously. To find them, substitute one equation into the other and solve for the variables, then verify the solutions by plugging them back into both original equations.