The three lines x-y+1=0 ,2x+y-4=0 ,and x+y+5=0 intersect to form a triangle. Find the coordinates of the vertices of the triangle.
x-y+1=0 ,2x+y-4=0 meet at (1,2)
2x+y-4=0 , x+y+5=0 meet at (9,-14)
x-y+1=0 , x+y+5=0 meet at (-3,-2)
ta-da!
To find the coordinates of the vertices of the triangle formed by the given lines, we need to find the points where these lines intersect.
Step 1: Solve the system of equations formed by the three given lines:
x - y + 1 = 0 ...(1)
2x + y - 4 = 0 ...(2)
x + y + 5 = 0 ...(3)
To solve this system, we can use any method, such as substitution or elimination. Let's use the elimination method here:
First, let's add (1) and (3) to eliminate the variable y:
x - y + 1 + x + y + 5 = 0
2x + 6 = 0
2x = -6
x = -3
Now, substitute the value of x back into equation (1) to solve for y:
-3 - y + 1 = 0
-2 - y = 0
y = -2
So, the first intersection point is (-3, -2).
Next, substitute the values of x and y into equation (2) and solve for the remaining variable:
2x + y - 4 = 0
2(-3) + (-2) - 4 = 0
-6 - 2 - 4 = 0
-12 = 0
This equation is not satisfied for any value of x and y. Hence, lines (2) and (3) do not intersect.
Therefore, the triangle formed by the given lines has only one vertex, which is (-3, -2).