## Solving a System of Equations: Circle and Line

This article will explore how to solve a system of equations consisting of a circle and a line. We'll delve into the process of finding the points of intersection, where the circle and line meet.

### Understanding the Equations

The first equation, **(x-2)^2 + (y-3)^2 = 16**, represents a circle. Here's why:

**Standard Form:**This equation is in the standard form of a circle: (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center and r is the radius.**Center and Radius:**From our equation, we can identify the center as (2, 3) and the radius as 4 (since the square root of 16 is 4).

The second equation, **x + y - 1 = 0**, represents a straight line.

**Slope-Intercept Form:**We can rewrite this equation in slope-intercept form (y = mx + b) as y = -x + 1. This tells us the slope is -1 and the y-intercept is 1.

### Finding the Points of Intersection

To find the points where the circle and line intersect, we need to solve for the values of x and y that satisfy *both* equations. Here's a common approach:

**Substitution Method:**Solve the linear equation for one variable (let's solve for x): x = 1 - y**Substitute:**Substitute this expression for x into the circle equation: (1 - y - 2)^2 + (y - 3)^2 = 16**Simplify and Solve:**This will give you a quadratic equation in terms of y. Solve this equation to find the values of y.**Back-Substitute:**Substitute the values of y you found back into either the original linear equation or the expression for x (x = 1 - y) to find the corresponding values of x.

### Graphical Interpretation

Visualizing the circle and line on a graph can provide a better understanding of the solution. The points of intersection represent the points where the two shapes touch. Depending on the relative positions of the circle and line, there could be:

**Two points of intersection:**The line intersects the circle at two distinct points.**One point of intersection:**The line is tangent to the circle, touching it at a single point.**No points of intersection:**The line does not intersect the circle.

### Conclusion

Solving a system of equations involving a circle and a line is a common problem in algebra and geometry. By using methods like substitution and understanding the shapes involved, we can find the points where these two figures intersect. Remember to check your solutions by plugging them back into the original equations.