The vertices of a triangle are the points of intersection of the line y=-x-1, x=2 and y = 1/5x + 13/5. Find an equation of the circle passing through all three vertices.

I don't understand how to solve this: should I set them all equal to find the vertices? But afterwards, then what?

First get the coordinates of the three vertices. It will help to plot the points.

The first two lines meet at x=2,y=-3
The second pair of lines meet at x=2, y = 2/5 +13/5 = 3
The first and third lines meet at
-x-1 = x/5 + 13/5
6x/5 = -18/5
x = -3, y = 2

The center of the circle through these points is located where two perpendicular side bisectors meet. Two side bisectors are y=0 and x = y. They meet at the origin: (0,0)

The radius of the circle is the distance from the origin to any vertex. That would be sqrt(4+9) = sqrt13

So the circle equation that passes through the three vertices is
x^2 + y^2 = 13

thanks a lot

To find the vertices of the triangle, you need to solve the given equations simultaneously. By setting the equations equal, you can find the points of intersection.

1. Start by setting the equations y = -x - 1 and x = 2 equal to each other:
-x - 1 = 2

2. Solve for x:
-x = 3
x = -3

3. Substitute the value of x back into either of the original equations to find y:
y = -(-3) - 1
= 3 - 1
= 2

So, you have the first vertex: (-3, 2).

Next, you will find the intersection point of the lines y = -x-1 and y = 1/5x + 13/5.

1. Set the two equations equal to each other:
-x - 1 = 1/5x + 13/5

2. Simplify the equation by multiplying both sides by 5:
-5x - 5 = x + 13

3. Combine like terms:
-5x - x = 13 + 5
-6x = 18

4. Divide both sides by -6:
x = -3

5. Substitute the value of x back into either of the equations to find y:
y = -(-3) - 1
= 3 - 1
= 2

So, you have the second vertex: (-3, 2).

Now, you have two vertices of the triangle, which are (-3, 2) and (2, y). You need to find the value of y to determine the third vertex.

Substitute x = 2 into the equation y = 1/5x + 13/5:
y = 1/5(2) + 13/5
= 2/5 + 13/5
= 15/5
= 3

So, the third vertex is (2, 3).

To find the equation of the circle passing through all three vertices, you can use the formula for the general equation of a circle:
(x - h)^2 + (y - k)^2 = r^2

Since the circle passes through the three vertices, you can substitute these values into the equation:

1. Using vertex (-3, 2):
(-3 - h)^2 + (2 - k)^2 = r^2

2. Using vertex (2, 3):
(2 - h)^2 + (3 - k)^2 = r^2

3. Using the third vertex (2, 3) as well:
(2 - h)^2 + (3 - k)^2 = r^2

Now, you have three equations with three variables: h, k, and r. You can solve this system of equations to find the equation of the circle passing through the three vertices.