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 using calculation. Then, check your answer by graphing them
To find the point of intersection of the two lines, we first need to find the equations of the two lines.
1. Line passing through points (3,0) and (0,2):
The slope of this line can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
m = (2 - 0) / (0 - 3) = 2 / -3 = -2/3
Using the point-slope form of the equation of a line:
y - y1 = m(x - x1)
y - 0 = (-2/3)(x - 3)
y = (-2/3)x + 2
2. Line passing through point (-2,1) with slope -3:
Using the point-slope form of the equation of a line:
y - y1 = m(x - x1)
y - 1 = -3(x + 2)
y - 1 = -3x - 6
y = -3x - 5
Now, we need to find the point of intersection by solving for x and y:
(-2/3)x + 2 = -3x - 5
(-2/3)x + 3x = -5 - 2
(7/3)x = -7
x = -3
Substitute x back into one of the equations to find y:
y = -3(-3) - 5
y = 9 - 5
y = 4
So, the point of intersection is (-3,4).
Now, let's check our answer by graphing the two lines:
Graph of y = (-2/3)x + 2 (line passing through points (3,0) and (0,2)) in blue.
Graph of y = -3x - 5 (line passing through point (-2,1) with slope -3) in red.
The point of intersection on the graph is at (-3,4), which confirms our calculation.