find the point of intersection of the line passing through points (3,0) and (0,2) and the line passing through point (-2,1) with slope -3 using calculation
To find the point of intersection, we first need to find the equations of the two lines.
1. Line passing through points (3,0) and (0,2):
Let the equation of the line be y = mx + b. We can find the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
m = (2 - 0) / (0 - 3)
m = 2 / -3
m = -2/3
Substitute either of the given points into the equation to solve for b:
0 = (-2/3)(3) + b
0 = -2 + b
b = 2
Therefore, the equation of the line passing through points (3,0) and (0,2) is y = (-2/3)x + 2.
2. Line passing through point (-2,1) with slope -3:
Using the point-slope form of the line equation:
y - y1 = m(x - x1)
Substitute the given point and slope to get:
y - 1 = -3(x + 2)
y - 1 = -3x - 6
y = -3x - 5
Now we can set the two equations equal to each other and solve for x:
(-2/3)x + 2 = -3x - 5
Add 3x to both sides and subtract 2 from both sides:
(3 - 2/3)x = -7
(9/3 - 2/3)x = -7
(7/3)x = -7
x = -7 * 3 / 7
x = -3
Substitute x back into one of the equations to solve for y:
y = -3(-3) - 5
y = 9 - 5
y = 4
Therefore, the point of intersection of the two lines is (-3, 4).