the graphs of the three equations below form a triangle. find the coordinates of the triangle's vertices.
4x+y=1
2x-y=5
3y+3x=12
solve the equations in pairs:
4x+y=1
2x-y=5
intersect at (1,-3)
Now do the same for the other pairs:
4x+y=1
3y+3x=12
2x-y=5
3y+3x=12
To find the coordinates of the triangle's vertices, we need to solve the system of equations. Let's solve the equations step by step.
1) 4x + y = 1
We can rewrite this equation in terms of y:
y = 1 - 4x
2) 2x - y = 5
Rewriting this equation in terms of y:
y = 2x - 5
3) 3y + 3x = 12
Let's divide both sides by 3 to simplify the equation:
y + x = 4
y = 4 - x
Now we have two equations for y in terms of x. By setting these two equations equal to each other, we can find the x-coordinate of one vertex of the triangle.
1 - 4x = 2x - 5
Moving the variables to one side:
6x = 6
Solving for x:
x = 1
To find the y-coordinate, we substitute x = 1 into either of the initial equations:
y = 1 - 4(1) = 1 - 4 = -3
So one vertex of the triangle is (1, -3).
Now, we can repeat this process for the other two pairs of equations.
For equations 1 and 3:
1 - 4x = 4 - x
Simplifying:
3x = 3
Solving for x:
x = 1
Substituting x = 1 into the initial equation:
y = 1 - 4(1) = 1 - 4 = -3
So the second vertex of the triangle is (1, -3).
For equations 2 and 3:
2x - 5 = 4 - x
Moving the variables to one side:
3x = 9
Solving for x:
x = 3
Substituting x = 3 into the initial equation:
y = 2(3) - 5 = 6 - 5 = 1
So the third vertex of the triangle is (3, 1).
Therefore, the coordinates of the triangle's vertices are (1, -3), (1, -3), and (3, 1).
To find the coordinates of the triangle's vertices, we need to solve the system of equations and find the intersection points. These intersection points will represent the vertices of the triangle.
Let's solve the system of equations step by step:
Equation 1: 4x + y = 1
Equation 2: 2x - y = 5
Equation 3: 3y + 3x = 12
We can use the method of substitution or elimination to solve this system. Let's use the method of substitution:
1. Solve Equation 2 for x:
2x - y = 5
2x = y + 5
x = (y + 5) / 2
2. Substitute x into Equation 1:
4((y + 5) / 2) + y = 1
(2y + 10) + y = 1
3y + 10 = 1
3y = 1 - 10
3y = -9
y = -3
3. Substitute y back into Equation 2 to find x:
2x - (-3) = 5
2x + 3 = 5
2x = 5 - 3
2x = 2
x = 1
So, one vertex of the triangle is (1, -3).
4. Substitute y into Equation 3 to find x:
3(-3) + 3x = 12
-9 + 3x = 12
3x = 12 + 9
3x = 21
x = 7
So, the second vertex of the triangle is (7, -3).
5. Substitute x into Equation 1 to find y:
4(7) + y = 1
28 + y = 1
y = 1 - 28
y = -27
So, the third vertex of the triangle is (7, -27).
Therefore, the coordinates of the triangle's vertices are (1, -3), (7, -3), and (7, -27).