does the line through (3,7) and (-2,-5) intersect with the line through (4,8) and (10.-2)
To determine if the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2), we can calculate the equations for both lines and check if their slopes are different.
1. Calculate the slope of the first line:
- Let (x1, y1) = (3, 7) and (x2, y2) = (-2, -5)
- The slope (m1) of the first line can be calculated using the formula: m1 = (y2 - y1) / (x2 - x1)
- Substitute the values: m1 = (-5 - 7) / (-2 - 3) = -12 / -5 = 12/5
2. Calculate the slope of the second line:
- Let (x1, y1) = (4, 8) and (x2, y2) = (10, -2)
- The slope (m2) of the second line can be calculated using the formula: m2 = (y2 - y1) / (x2 - x1)
- Substitute the values: m2 = (-2 - 8) / (10 - 4) = -10 / 6 = -5/3
3. Check if the slopes are different:
- If the two lines intersect, their slopes should be different.
- Comparing the slopes obtained, m1 = 12/5 and m2 = -5/3, we can conclude that the two lines intersect.
Therefore, the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2).
To determine if the two lines intersect, we need to find the equations of the lines and see if there is a common point.
The equation of a line can be found using the slope-intercept form, which is given by:
y = mx + b
where m is the slope and b is the y-intercept.
First, let's find the slope of the line passing through (3, 7) and (-2, -5). The slope (m1) is given by:
m1 = (y2 - y1) / (x2 - x1)
= (-5 - 7) / (-2 - 3)
= -12 / (-5)
= 12/5
Now, let's find the slope of the line passing through (4, 8) and (10, -2). The slope (m2) is given by:
m2 = (y2 - y1) / (x2 - x1)
= (-2 - 8) / (10 - 4)
= -10 / 6
= -5/3
The equation of the line passing through (3, 7) and (-2, -5) is:
y = (12/5)x + b1
To find b1, substitute the coordinates of any point on the line (let's take (3, 7)) into the equation:
7 = (12/5)(3) + b1
35 = 36 + 5b1
-1 = 5b1
b1 = -1/5
So, the equation of the line passing through (3, 7) and (-2, -5) is:
y = (12/5)x - 1/5
The equation of the line passing through (4, 8) and (10, -2) is:
y = (-5/3)x + b2
To find b2, substitute the coordinates of any point on the line (let's take (4, 8)) into the equation:
8 = (-5/3)(4) + b2
24 = -20 + 15b2
44 = 15b2
b2 = 44/15
So, the equation of the line passing through (4, 8) and (10, -2) is:
y = (-5/3)x + 44/15
Now let's check if the two lines intersect by solving the system of equations formed by setting the two equations equal to each other:
(12/5)x - 1/5 = (-5/3)x + 44/15
Multiplying through by 15 to eliminate the fractions:
36x - 3 = -25x + 44
Combining like terms:
61x = 47
x = 47/61
Now substitute this value of x back into either of the equations to find the corresponding y-coordinate. Using the first equation:
y = (12/5)(47/61) - 1/5
y = 564/305 - 61/305
y = 503/305
Therefore, the lines intersect at the point (47/61, 503/305).
To determine whether the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2), we can use the concept of slope-intercept form.
1. Find the slope of the first line:
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula,
m = (y2 - y1) / (x2 - x1)
Let's calculate the slope of the first line:
m1 = (-5 - 7) / (-2 - 3)
= (-12) / (-5)
= 12/5
2. Find the slope of the second line:
Similarly, calculate the slope of the second line:
m2 = (-2 - 8) / (10 - 4)
= (-10) / 6
= -5/3
3. Check if the slopes are equal:
If the slopes of two lines are equal, they are parallel and will never intersect. If they are not equal, they might intersect.
Let's compare the slopes:
m1 = 12/5 and m2 = -5/3
Since the slopes are not equal, the two lines are not parallel and could intersect.
4. Find the y-intercept of each line:
To determine the equations of the lines, we need to find the y-intercepts (b) using the point-slope form (y = mx + b).
For the first line (3,7) and (-2,-5):
Using the point-slope form, we can choose either point and substitute the coordinates:
7 = (12/5) * 3 + b
Solving for b:
7 = 36/5 + b
Multiply through by 5 to eliminate the denominator:
35 = 36 + 5b
5b = -1
b = -1/5
The equation of the first line is y = (12/5)x - 1/5.
For the second line (4,8) and (10,-2):
8 = (-5/3) * 4 + b
Solving for b:
8 = -20/3 + b
Multiply through by 3 to eliminate the denominator:
24 = -20 + 3b
3b = 44
b = 44/3
The equation of the second line is y = (-5/3)x + 44/3.
5. Check if the lines intersect:
To find out if the lines intersect, we need to check if their equations have a common solution (x, y).
Let's solve the system of equations:
(12/5)x - 1/5 = (-5/3)x + 44/3
Multiply through by 15 to eliminate the denominators:
36x - 3 = -25x + 220
Move the variables to one side and constants to the other:
36x + 25x = 220 + 3
61x = 223
x = 223/61
Substitute x back into either equation to find y:
y = (12/5)*(223/61) - 1/5
y = 67/61
Therefore, the lines intersect at the point (223/61, 67/61).
So, the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2) at the point (223/61, 67/61).
does the line through (3,7) and (-2,-5) intersect with the line through (4,8) and (10.-2)
why