Does the line through (3,7) and (-2,-5) intersect with the line through (4,8)-‘d (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 find the equations of both lines and see if they intersect.
First, let's find the equation of the line through (3,7) and (-2,-5):
Using the formula for the equation of a line (y - y1) = m(x - x1), where m is the slope of the line and (x1,y1) is a point on the line, we can calculate the slope (m).
m = (y2 - y1) / (x2 - x1)
m = (-5 - 7) / (-2 - 3)
m = -12 / -5
m = 12/5
Now, substituting the slope and one of the points (3,7) into the equation:
(y - 7) = (12/5)(x - 3)
Expanding the equation:
y - 7 = (12/5)x - 36/5
Bringing all terms to the left-hand side:
(12/5)x - y = -36/5 + 7
(12/5)x - y = -36/5 + 35/5
(12/5)x - y = -1/5
So, the equation of the line through (3,7) and (-2,-5) is (12/5)x - y = -1/5.
Next, let's find the equation of the line through (4,8) and (10,-2):
Using the formula for the equation of a line (y - y1) = m(x - x1), we can calculate the slope (m).
m = (y2 - y1) / (x2 - x1)
m = (-2 - 8) / (10 - 4)
m = -10 / 6
m = -5/3
Now, substituting the slope and one of the points (4,8) into the equation:
(y - 8) = (-5/3)(x - 4)
Expanding the equation:
y - 8 = (-5/3)x + 20/3
Bringing all terms to the left-hand side:
(-5/3)x + y = 20/3 + 8
(-5/3)x + y = 20/3 + 24/3
(-5/3)x + y = 44/3
So, the equation of the line through (4,8) and (10,-2) is (-5/3)x + y = 44/3.
To check if the two lines intersect, we can compare the coefficients of x and y of the two equations. If the coefficients of x and y are different, then the lines intersect.
Comparing the coefficients of x and y:
(12/5)x - y = -1/5
(-5/3)x + y = 44/3
The coefficients of x (-5/3 and 12/5) and y (1 and -1) are different, indicating that the lines are not parallel and therefore must intersect.
Hence, the line through (3,7) and (-2,-5) intersects 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), you can use the slope-intercept form of a linear equation.
1. Find the slope of the line through (3,7) and (-2,-5):
The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Plugging in the values:
m = (-5 - 7) / (-2 - 3)
m = (-12) / (-5)
m = 12/5
2. Find the equation of the line passing through (3,7) using the point-slope form:
The equation of a line with slope (m) passing through a point (x1, y1) is given by the formula:
y - y1 = m * (x - x1)
Plugging in the values:
y - 7 = (12/5) * (x - 3)
Simplifying:
5y - 35 = 12x - 36
5y = 12x - 1
12x - 5y = 1
3. Find the slope of the line through (4,8) and (10,-2):
Using the same formula as above:
m = (-2 - 8) / (10 - 4)
m = (-10) / 6
m = -5/3
4. Find the equation of the line passing through (4,8) using the point-slope form:
Following the same steps as before:
y - 8 = (-5/3) * (x - 4)
Simplifying:
3y - 24 = -5x + 20
5x + 3y = 44
Now, you can compare the equations of the two lines:
12x - 5y = 1 and 5x + 3y = 44
If the lines intersect, there should be a unique solution (x, y) that satisfies both equations. You can solve this system of equations using various methods such as substitution or elimination to confirm if they intersect or not.
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 of both lines and see if they have a common point.
1. First, let's find the equation of the line through (3,7) and (-2,-5):
- Begin by calculating the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (-5 - 7) / (-2 - 3)
m = (-12) / (-5)
m = 12/5
- Now, we can use the point-slope form to write the equation of the line:
y - y1 = m(x - x1)
y - 7 = (12/5)(x - 3)
y - 7 = (12/5)x - 36/5
- Rearrange the equation to standard form:
(12/5)x - y = 36/5 - 7
(12/5)x - y = 36/5 - 35/5
(12/5)x - y = 1/5
So, the equation of the first line is (12/5)x - y = 1/5.
2. Next, let's find the equation of the line through (4,8) and (10,-2):
- Calculate the slope (m):
m = (y2 - y1) / (x2 - x1)
m = (-2 - 8) / (10 - 4)
m = (-10) / 6
m = -5/3
- Use the point-slope form with either of the given points:
y - y1 = m(x - x1)
y - 8 = (-5/3)(x - 4)
y - 8 = (-5/3)x + (20/3)
- Rearrange the equation to standard form:
(-5/3)x + y = 20/3 + 8
(-5/3)x + y = 20/3 + 24/3
(-5/3)x + y = 44/3
So, the equation of the second line is (-5/3)x + y = 44/3.
Now, we need to check if these two lines intersect or have a common point. We can do this by comparing their slopes. If the slopes are different, the lines intersect. If the slopes are the same, the lines are parallel and do not intersect.
In this case, the first line has a slope of 12/5 and the second line has a slope of -5/3. Since these slopes are different, the lines intersect.
Therefore, the line through (3,7) and (-2,-5) intersects with the line through (4,8) and (10,-2).