Does the line through (3,7) and (-2,-5) intersect 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 using the given points and then check if they have a common solution.
Let's find the equation of the first line through the points (3,7) and (-2,-5).
First, determine the slope of the line (m1) using the formula:
m1 = (y2 - y1) / (x2 - x1)
m1 = (-5 - 7) / (-2 - 3)
m1 = -12 / -5
m1 = 12/5
Now, let's use the point-slope form of a linear equation (y - y1) = m (x - x1) to find the equation of the line:
(y - 7) = (12/5)(x - 3)
5(y - 7) = 12(x - 3)
5y - 35 = 12x - 36
12x - 5y = 1
This is the equation of the first line.
Next, let's find the equation of the second line through the points (4,8) and (10,-2).
Similar to the previous steps, first determine the slope of the line (m2):
m2 = (y2 - y1) / (x2 - x1)
m2 = (-2 - 8) / (10 - 4)
m2 = -10 / 6
m2 = -5/3
Using the point-slope form, we have:
(y - 8) = (-5/3)(x - 4)
3(y - 8) = -5(x - 4)
3y - 24 = -5x + 20
5x + 3y = 44
This is the equation of the second line.
To check if the two lines intersect, we need to solve the system of equations:
12x - 5y = 1
5x + 3y = 44
We can solve this system by either substitution or elimination method.
Using the elimination method, multiply the first equation by 3 and the second equation by 5 to eliminate the variable y:
36x - 15y = 3
25x + 15y = 220
Summing the two equations, we get:
61x = 223
x = 223 / 61 ≈ 3.66
Substituting x back into the second equation:
5(3.66) + 3y = 44
18.3 + 3y = 44
3y = 44 - 18.3
3y = 25.7
y = 25.7 / 3 ≈ 8.57
Therefore, the lines intersect at approximately (3.66, 8.57).
To determine if the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2), we need to find the equations of the two lines and check if they intersect.
1. Finding the equation of the first line:
Let's use the point-slope form of a linear equation: y - y1 = m(x - x1).
Let (x1, y1) = (3, 7), and let's find the slope (m):
m = (y2 - y1) / (x2 - x1)
= (-5 - 7) / (-2 - 3)
= -12 / -5
= 12/5
Now substitute the slope (m) and a point (x1, y1) into the equation:
y - 7 = (12/5)(x - 3)
2. Finding the equation of the second line:
Similarly, using the point-slope form:
Let (x1, y1) = (4, 8), and let (x2, y2) = (10, -2).
Find the slope (m) using:
m = (y2 - y1) / (x2 - x1)
= (-2 - 8) / (10 - 4)
= -10 / 6
= -5/3
Substitute the slope (m) and a point (x1, y1) into the equation:
y - 8 = (-5/3)(x - 4)
3. Checking if the lines intersect:
To see if the two lines intersect, we need to find the point (x, y) that satisfies both equations.
So, we can set the two equations equal to each other and solve for x and y:
(12/5)(x - 3) + 7 = (-5/3)(x - 4) + 8
Multiply both sides by 15 to clear the fractions:
36(x - 3) + 105 = -25(x - 4) + 120
Distribute:
36x - 108 + 105 = -25x + 100 + 120
Combine like terms:
61x - 3 = -25x + 220
Add 25x to both sides, and subtract 3 from both sides:
86x = 223
Divide both sides by 86:
x = 223/86
Substitute x back into either of the original equations to solve for y.
Therefore, the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2) at the point (223/86, y).
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 slope-intercept form of a linear equation and compare the slopes of the two lines.
Step 1: Find the equation of the first line.
The equation of a line in slope-intercept form (y = mx + b) can be found using the formula:
m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
For the first line passing through (3,7) and (-2,-5), the slope (m1) can be calculated as follows:
m1 = (-5 - 7) / (-2 - 3)
= (-12) / (-5)
= 12/5
Using the point-slope form, we can substitute one of the given points (3,7) and the calculated slope (12/5) to find the equation of the first line:
y - 7 = (12/5)(x - 3)
5y - 35 = 12x - 36
12x - 5y = -1
So, the equation of the first line is 12x - 5y = -1.
Step 2: Find the equation of the second line.
Using the same process, we can find the equation of the second line passing through (4,8) and (10,-2).
The slope (m2) can be calculated as:
m2 = (-2 - 8) / (10 - 4)
= (-10) / (6)
= -5/3
Using the point-slope form and substituting the point (4,8) and the slope (-5/3):
y - 8 = (-5/3)(x - 4)
3y - 24 = -5x + 20
5x + 3y = 44
So, the equation of the second line is 5x + 3y = 44.
Step 3: Compare the slopes.
Since the slopes of the two lines (m1 = 12/5 and m2 = -5/3) are not the same, the lines are not parallel.
Step 4: Check for intersection.
To check if the lines intersect, we can solve the two equations simultaneously by finding the values of x and y that satisfy both equations.
Solving the equations 12x - 5y = -1 and 5x + 3y = 44, we can find the values of x and y.
After solving the system of equations, we find that x = 4 and y = -2.
Since there is a solution for x and y that satisfies both equations, the lines do intersect.
Therefore, the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2).