Let's solve the given system of equations and determine which graph matches:

The equations are:

1. $x^2 + y^2 = 25$ (a circle centered at the origin with radius $5$)
2. $x + y = 3$ (a straight line with slope $-1$ and y-intercept $3$).

Step 1: Graphing the Circle

The circle equation $x^2 + y^2 = 25$ represents a circle centered at the origin $(0, 0)$ with radius $5$. Its points are all at a distance of $5$ from the origin.

Step 2: Graphing the Line

The line equation $x + y = 3$ can be rewritten as:
$y = -x + 3.$
This line has a slope of $-1$ and a y-intercept of $3$. It crosses the y-axis at $(0, 3)$ and the x-axis at $(3, 0)$.

Step 3: Solving for Intersection Points

We solve for the points where the circle and the line intersect by substituting $y = -x + 3$ into $x^2 + y^2 = 25$:
$x^2 + (-x + 3)^2 = 25$ Expanding $(-x + 3)^2$:
$x^2 + (x^2 - 6x + 9) = 25$ Combine like terms:
$2x^2 - 6x + 9 = 25$ Simplify:
$2x^2 - 6x - 16 = 0$ Divide by 2:
$x^2 - 3x - 8 = 0$ Solve using the quadratic formula:
$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-8)}}{2(1)}$ $x = \frac{3 \pm \sqrt{9 + 32}}{2}$ $x = \frac{3 \pm \sqrt{41}}{2}$ The solutions for $x$ are approximately:
$x = \frac{3 + \sqrt{41}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{41}}{2}.$

Substitute these $x$-values into $y = -x + 3$ to find the corresponding $y$-values.

Step 4: Choosing the Correct Graph

The circle and the line intersect at two points, and the line is tangent to the circle's curvature. Based on the options:

• Graph C is the correct choice, as it shows the line intersecting the circle at exactly two points.

Would you like further explanation on any step?

Related Questions:

1. How do we derive the equation of a line from a given slope and point?
2. How do we solve quadratic equations with complex roots?
3. What are the steps to sketch the graph of a circle from its equation?
4. How do you find the points of intersection between two curves?
5. How does changing the radius affect the equation of a circle?

Tip: Always check for symmetry in equations involving circles to simplify graphing!