Line A passes through points (-1, 5) and (3, -1). Line B passes through points (7, 2) and (6, -1).
At what point does line A intersect line B?
To find the intersection point of two lines, we need to set their equations equal to each other and solve for the variables. First, let's find the equation of line A.
The slope of line A can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates (-1, 5) and (3, -1):
m = (-1 - 5) / (3 - (-1))
m = (-6) / (4)
m = -3/2
Using the point-slope form of a linear equation, the equation of line A becomes:
y - y1 = m(x - x1)
y - 5 = (-3/2)(x - (-1))
y - 5 = (-3/2)(x + 1)
Expanding and rearranging the equation:
y - 5 = (-3/2)x - 3/2
y = (-3/2)x - 3/2 + 5
y = (-3/2)x - 3/2 + 10/2
y = (-3/2)x + 7/2
The equation of line A is y = (-3/2)x + 7/2.
Now, let's find the equation of line B using the same process.
The slope of line B can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the coordinates (7, 2) and (6, -1):
m = (-1 - 2) / (6 - 7)
m = (-3) / (-1)
m = 3
Using the point-slope form of a linear equation, the equation of line B becomes:
y - y1 = m(x - x1)
y - 2 = 3(x - 7)
y - 2 = 3x - 21
Rearranging the equation:
y = 3x - 21 + 2
y = 3x - 19
The equation of line B is y = 3x - 19.
To find the intersection point, we can set the two equations equal to each other:
(-3/2)x + 7/2 = 3x - 19
Multiplying the equation by 2 to eliminate the fractions:
-3x + 7 = 6x - 38
Adding 3x to both sides and subtracting 7 from both sides:
0 = 9x - 45
Simplifying:
9x = 45
Dividing both sides by 9:
x = 5
Substituting x = 5 into one of the equations (let's use the equation of line A):
y = (-3/2)x + 7/2
y = (-3/2)(5) + 7/2
y = -15/2 + 7/2
y = -8/2
y = -4
Therefore, the intersection point of line A and line B is (5, -4).
To find the point where line A intersects line B, we need to first find the equations of both lines.
The equation of a line is given by the formula: y = mx + b, where m represents the slope of the line, and b represents the y-intercept.
Let's find the equation of line A:
Step 1: Find the slope of line A
m = (y2 - y1) / (x2 - x1) (Using the points (-1, 5) and (3, -1))
= (-1 - 5) / (3 - (-1))
= (-6) / (4)
= -3/2
Step 2: Find the y-intercept (b) of line A
Using the formula: y = mx + b, we can substitute any point from line A and the slope we just found to find the value of b. Let's use the point (-1,5).
5 = (-3/2)(-1) + b
5 = 3/2 + b
5 - 3/2 = b
10/2 - 3/2 = b
7/2 = b
So, the equation of line A is y = -3/2x + 7/2
Now let's find the equation of line B:
Step 1: Find the slope of line B
m = (y2 - y1) / (x2 - x1) (Using the points (7, 2) and (6, -1))
= (-1 - 2) / (6 - 7)
= (-3) / (-1)
= 3
Step 2: Find the y-intercept (b) of line B
Using the formula: y = mx + b, we can substitute any point from line B and the slope we just found to find the value of b. Let's use the point (6, -1).
-1 = 3(6) + b
-1 = 18 + b
-19 = b
So, the equation of line B is y = 3x - 19
Now we can find the point of intersection by setting the two equations equal to each other:
-3/2x + 7/2 = 3x - 19
Now, we can solve for x:
-3/2x - 3x = -19 - 7/2
-9/2x = -45/2 - 7/2
-9/2x = -52/2
x = (-52/2) / (-9/2)
x = (-52/2) * (-2/9)
x = 104/18
x = 5.778
Then we substitute the value of x back into either of the original equations to find the value of y:
y = 3(5.778) - 19
y = 17.334 - 19
y = -1.666
Therefore, the point of intersection is approximately (5.778, -1.666).