Below are two pairs of points. A line goes through each pair of points. What are the coordinates for the intersection of the two lines? Show your work. Pair 1 (1,1) and (7,7)
Pair 2 (0,8) and (8,0)
first line:
slope = (7-1)/(7-1) = 1
y = 1x + b
using either point, I will use the simpler (1,1)
1 = 1 + b
b = 0
first equation: y = x
2nd line:
follow my steps....
to solve, sub in y = x from the 1st into the 2nd
To find the intersection point of two lines, we need to first find the equations of the lines using the given pairs of points. Then, we can solve these equations simultaneously to find the coordinates of the intersection point.
Let's find the equation of the line that passes through the points (1,1) and (7,7) first.
1. Find the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1)
Here, (x1, y1) = (1,1) and (x2, y2) = (7,7)
m = (7 - 1) / (7 - 1) = 6 / 6 = 1
2. Use the point-slope form of a line (y - y1) = m(x - x1), and substitute the coordinates of one of the given points. Let's use (1,1).
y - 1 = 1(x - 1)
y - 1 = x - 1
y = x
The equation of the line passing through (1,1) and (7,7) is y = x.
Now, let's find the equation of the line that passes through the points (0,8) and (8,0).
1. Find the slope (m) of the line using the formula: m = (y2 - y1) / (x2 - x1)
Here, (x1, y1) = (0,8) and (x2, y2) = (8,0)
m = (0 - 8) / (8 - 0) = -8 / 8 = -1
2. Use the point-slope form of a line (y - y1) = m(x - x1), and substitute the coordinates of one of the given points. Let's use (0,8).
y - 8 = -1(x - 0)
y - 8 = -x
y = -x + 8
The equation of the line passing through (0,8) and (8,0) is y = -x + 8.
Now, to find the intersection point of the two lines, we can solve the system of equations: y = x and y = -x + 8.
Substitute the second equation into the first equation to eliminate y:
x = -x + 8
Now, solve for x:
2x = 8
x = 8 / 2
x = 4
Substitute x = 4 back into the first equation to find y:
y = x
y = 4
Therefore, the coordinates of the intersection point are (4, 4).