Question

find the point of intersection of the line passing through two points (3,0) and (0,2) and the line passing through point (-2,1) with slope -3

Answer 1

First, we need to find the equation of the line passing through points (3,0) and (0,2).

The slope of the line passing through these two points can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
m = (2 - 0) / (0 - 3)
m = 2 / -3
m = -2/3

So, the equation of the line passing through (3,0) and (0,2) is:
y = -2/3x + b

Now, we can substitute the values of one of the points (3,0) into the equation to find the value of b:
0 = -2/3(3) + b
0 = -2 + b
b = 2

Therefore, the equation of the line passing through (3,0) and (0,2) is:
y = -2/3x + 2

Next, we need to find the equation of the line passing through the point (-2,1) with slope -3. The equation of the line with slope m passing through point (x1, y1) is given by:
y - y1 = m(x - x1)

Substitute the values into the equation:
y - 1 = -3(x + 2)
y - 1 = -3x - 6
y = -3x - 5

Now, we need to find the point of intersection of the two lines by solving the system of equations:
y = -2/3x + 2
y = -3x - 5

Substitute the second equation into the first:
-3x - 5 = -2/3x + 2
-9x - 15 = -2x + 6
-7x = 21
x = -3

Substitute x back into the second equation to find y:
y = -3(-3) - 5
y = 9 - 5
y = 4

Therefore, the point of intersection of the two lines is (-3, 4).