Answer: a) x - y = -1
b) 3x + 2y = 13
Explanation:
a) To find the equation of the line passing through points P(2, 3) and Q(5, 6), we can use the point-slope form of the equation of a line, which is:
y - y1 = m(x - x1)
where m is the slope of the line, and (x1, y1) is one of the points on the line.
First, we need to find the slope of the line passing through P and Q. The slope, m, is given by:
m = (y2 - y1)/(x2 - x1)
where (x1, y1) = (2, 3) and (x2, y2) = (5, 6).
m = (6 - 3)/(5 - 2) = 1
Now that we have the slope, we can use the point-slope form of the equation of a line to find the equation of the line passing through P and Q:
y - 3 = 1(x - 2)
Simplifying the equation, we get:
y - 3 = x - 2
y = x + 1
To convert this equation to standard form, we can rearrange the terms to get:
x - y = -1
Therefore, the equation of the line passing through points P(2, 3) and Q(5, 6) in standard form is x - y = -1.
b) To find the equation of the line passing through points A(5, -1) and B(1, 5), we can use the same method as in part (a).
The slope of the line passing through A and B is:
m = (y2 - y1)/(x2 - x1)
where (x1, y1) = (5, -1) and (x2, y2) = (1, 5).
m = (5 - (-1))/(1 - 5) = -3/2
Using the point-slope form of the equation of a line, we get:
y - (-1) = (-3/2)(x - 5)
Simplifying the equation, we get:
y = (-3/2)x + 13/2
To convert this equation to standard form, we can rearrange the terms to get:
3x + 2y = 13
Therefore, the equation of the line passing through points A(5, -1) and B(1, 5) in standard form is 3x + 2y = 13.