Answer:
1. To find the gradient of a line, we can use the slope-intercept form of the equation: y = mx + b, where m is the gradient and b is the y-intercept.
a) To find the gradient of (x+7)/(y-2) = 0, we can first rewrite the equation as y = (2x+14)/(x+7). To find the slope, we can take the derivative of y with respect to x. We get the slope or the gradient of the line as m = 2/(x+7).
b) To find the gradient of the line through (p,5) and (6,2p), we can use the point-slope form of the equation: y - y1 = m(x - x1). We can substitute the coordinates of the two points and solve for m.
y - 5 = m(x - p)
2p - 5 = m(6 - p)
2p - 5 = 6m - mp
mp + 6m = 2p + 5
m(p+6) = 2p + 5
m = (2p+5)/(p+6)
2. To find the equation of a line, we can use the slope-intercept form of the equation: y = mx + b, where m is the gradient and b is the y-intercept.
a) To find the equation of the line with a gradient of 3, passing through (0,5), we can substitute the values into the slope-intercept form.
y = 3x + b
5 = 3(0) + b
b = 5
The equation of the line is y = 3x + 5
3. To find the value of p, if the gradient of the line joining (-1,p) and (p, 4) is 2/3, we can use the point-slope form of the equation: y - y1 = m(x - x1). We can substitute the coordinates of the two points and the gradient, and solve for p.
y - p = (2/3)(x - (-1))
4 - p = (2/3)(p - (-1))
4 - p = (2/3)p + (2/3)
(2/3)p - 4 + p = (2/3)
p = 6
Final Answer: The value of p is 6.