7) Decide whether or not the equation has a circle as its graph. \( x^{2}+y^{2}-8 x+8 y+23=0 \)

To determine if the equation represents a circle, we can first rewrite it in the standard form of a circle's equation, which is \((x - h)^2 + (y - k)^2 = r^2\). Start by rearranging and completing the square: 1. Rearranging the equation: \[ x^2 - 8x + y^2 + 8y + 23 = 0 \] \[ x^2 - 8x + y^2 + 8y = -23 \] 2. Completing the square: - For \(x\): \[ x^2 - 8x = (x - 4)^2 - 16 \] - For \(y\): \[ y^2 + 8y = (y + 4)^2 - 16 \] Now substitute back into the equation: \[ (x - 4)^2 - 16 + (y + 4)^2 - 16 = -23 \] \[ (x - 4)^2 + (y + 4)^2 - 32 = -23 \] \[ (x - 4)^2 + (y + 4)^2 = 9 \] This is indeed a circle with center \((4, -4)\) and radius \(3\). So, the original equation does have a circle as its graph! --- Did you know that the concept of a circle has been studied for thousands of years? Ancient Greek mathematicians like Euclid and Archimedes laid the groundwork for geometry, including circle properties, which are still essential in various fields today! Circles pop up everywhere in the real world! From wheels to pizza to the orbits of planets, understanding circles is foundational in engineering, astronomy, and even art. So, whether you're designing a bike tire or painting the perfect round canvas, the circle is your best friend!