At how many distinct points do any two of the three graphs x2 + y2 = 25, 4x + 3y = 25 and x = 0 meet each other?2354

Sure, let's solve this step by step:

1. The first equation represents a circle with radius 5 centered at the origin (0,0). The equation can be written as (x-0)² + (y-0)² = 5².

2. The second equation represents a straight line. We can rearrange it to the form y = mx + c, where m is the slope and c is the y-intercept. So, 3y = -4x + 25, which simplifies to y = -4/3x + 25/3.

3. The third equation represents a vertical line passing through the origin (0,0).

Now, let's find the intersection points:

4. The circle and the line intersect where both equations are satisfied. Substituting y from the line equation into the circle equation gives: x² + (-4/3x + 25/3)² = 25. Solving this quadratic equation will give the x-coordinates of the intersection points.

5. The circle and the vertical line intersect at x=0. Substituting x=0 into the circle equation gives y² = 25, so y = ±5. Therefore, the vertical line intersects the circle at (0,5) and (0,-5).

6. The line and the vertical line intersect at x=0. Substituting x=0 into the line equation gives y = 25/3. Therefore, the vertical line intersects the line at (0, 25/3).

So, there are 4 distinct intersection points: two where the circle and the line intersect, one where the circle and the vertical line intersect, and one where the line and the vertical line intersect.