does the line through 3,7 and -2,-5 intersect with the line through 4,8 and 10,-2
Yes, the lines intersect. The line passing through (3,7) and (-2,-5) has a slope of (7 - (-5))/(3 - (-2)) = 12/5.
The line passing through (4,8) and (10,-2) has a slope of (8 - (-2))/(4 - 10) = 10/(-6) = -5/3.
Since the slopes are different, the lines will intersect at a single point.
To determine if the line through the points (3,7) and (-2,-5) intersects with the line through the points (4,8) and (10,-2), we can use the slope-intercept form of the equation for a line (y = mx + b).
Step 1: Find the slope (m1) of the first line through the points (3,7) and (-2,-5).
m1 = (y2 - y1) / (x2 - x1)
= (-5 - 7) / (-2 - 3)
= (-12) / (-5)
= 12/5
Step 2: Find the slope (m2) of the second line through the points (4,8) and (10,-2).
m2 = (y2 - y1) / (x2 - x1)
= (-2 - 8) / (10 - 4)
= (-10) / 6
= -5/3
Step 3: If the two lines intersect, their slopes will be different. Let's check if the slopes are equal or not.
Since m1 = 12/5 and m2 = -5/3 are not equal, the two lines are not parallel.
Step 4: To determine if the two lines intersect, we can compare the y-intercepts (b-values) of the two lines.
For the first line, y = mx + b, where m = 12/5 and (x, y) = (3, 7):
7 = (12/5)(3) + b
b = 7 - (12/5)(3)
b = 7 - 36/5
b = 35/5 - 36/5
b = -1/5
For the second line, y = mx + b, where m = -5/3 and (x, y) = (4, 8):
8 = (-5/3)(4) + b
8 = -20/3 + b
b = 8 + 20/3
b = 24/3 + 20/3
b = 44/3
Step 5: If the y-intercepts are different, the two lines intersect. Let's compare the y-intercepts.
Since -1/5 and 44/3 are different, the two lines intersect.
Therefore, the line through the points (3,7) and (-2,-5) intersects with the line through the points (4,8) and (10,-2).
To determine if the two lines intersect, we can use the equation of a line in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
First, we need to find the slope (m) for both lines. The slope can be calculated using the formula: m = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
For the line through the points (3, 7) and (-2, -5), the slope (m1) is calculated as:
m1 = (-5 - 7)/(-2 - 3) = -12/(-5) = 12/5 = 2.4
Similarly, for the line through the points (4, 8) and (10, -2), the slope (m2) is:
m2 = (-2 - 8)/(10 - 4) = -10/6 = -5/3 ≈ -1.67
Now that we have the slopes, we can compare them. If the slopes are different, the lines will intersect at a single point. However, if the slopes are the same, the lines are parallel and do not intersect.
Since m1 = 2.4 and m2 = -1.67, the slopes are different. Therefore, the two lines will intersect.