Answer:
a) The equation of this line is y = -2x + 3
b) The equation of this line is y = -3x + 7
c) The equation of this line is y = -1/2x + 4 or y = -0.5x + 4
d) The equation of this line is y = 2
Explanation:
The equation of a line can be written in the form y = mx + c, where m is the gradient and c is the y-intercept.
a) The equation of the line with a gradient of -2 and cutting the y-axis at 3 can be found using the point-slope form of a linear equation. The point-slope form is given by y - y1 = m(x - x1), where m is the gradient and (x1, y1) is a point on the line. Substituting m = -2 and (x1, y1) = (0, 3), we get:
y - 3 = -2(x - 0)
Simplifying the right-hand side gives:
y - 3 = -2x
Adding 3 to both sides gives the final equation:
y = -2x + 3
b) The equation of the line with a gradient of -3 and passing through the point (2, 1) can be found using the point-slope form again. Substituting m = -3 and (x1, y1) = (2, 1), we get:
y - 1 = -3(x - 2)
Simplifying the right-hand side gives:
y - 1 = -3x + 6
Adding 1 to both sides gives the final equation:
y = -3x + 7
c) To find the equation of the line passing through the points (2, 3) and (4, 2), we first need to find its gradient. The gradient is given by:
m = (y2 - y1)/(x2 - x1)
Substituting the coordinates of the two points, we get:
m = (2 - 3)/(4 - 2) = -1/2 or -0.5
Now, we can use the point-slope form again, this time with (x1, y1) = (2, 3) and m = -1/2:
y - 3 = (-1/2)(x - 2)
Simplifying the right-hand side gives:
y - 3 = (-1/2)x + 1
Adding 3 to both sides gives the final equation:
y = (-1/2)x + 4 or y = -0.5x + 4
d)The line is parallel to the x-axis and passes through the point (-3 ; 2). A line parallel to the x-axis has a gradient of 0. The general equation of a line is y = mx + c, where m is the gradient and c is the y-intercept. Since the gradient is 0, the equation becomes y = c. Since the line passes through the point (-3 ; 2), we can substitute y = 2 into the equation to find that c = 2. Therefore, the equation of this line is y = 2.