Find the Union of the Inequalities x^2+y^2=100 , 8x-6y=0

Problem

x2+y2=100∪8*x−6*y=0

Solution

1. Identify the geometric shapes represented by the equations. The equation x2+y2=100 represents a circle centered at the origin (0,0) with a radius of 10

2. Identify the second equation. The equation 8*x−6*y=0 represents a line passing through the origin.

3. Simplify the linear equation by solving for y to put it in slope-intercept form.

8*x=6*y

y=8/6*x

y=4/3*x

4. Define the union. The union of two sets of points is the set of all points that satisfy at least one of the equations. In this context, the union is the collection of all points on the circle and all points on the line.

5. Express the final set. Since these are two distinct geometric objects, the union is simply the set of points (x,y) such that x2+y2=100 or y=4/3*x

Final Answer

(x2+y2=100)∪(8*x−6*y=0)={(x,y)∈ℝ2:x2+y2=100* or *y=4/3*x}

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