Final answer:
To find the equation of a line passing through two points, use the point-slope formula. For vertical lines, the equation is x = constant and for horizontal lines, the equation is y = constant. To find lines parallel and perpendicular, use the same formula with a different slope. To solve for y, rearrange the equation to isolate y.
Step-by-step explanation:
a) To determine the equation of a line passing through two points, we can use the point-slope formula. The formula is given by y - y1 = m(x - x1), where (x1, y1) are the coordinates of one of the points and m is the slope. Substituting the values into the formula, we can find the equation of the line. For example, for (4,0) and (1,2) the equation of the line is y = -2x + 8.
b) We can use the same method for (5,2) and (3,-1). The equation of the line is y = 3x - 7.
c) For (-1,3) and (-1,4), it's a vertical line passing through x = -1. Thus, the equation is x = -1.
d) For (2,4) and (-5,4), it's a horizontal line passing through y = 4. It can be represented as y = 4.
a) We can use the point-slope formula to find the equation of the line through (-2,-5) and (0,0). The equation is y = (5/2)x - 2, parallel line: y = (5/2)x + 2, perpendicular line: y = (-2/5)x - 1.
b) Using the point-slope formula again, the equation of the line through (1,2) and (-3,4) is y = (-1/2)x + 5/2, parallel line: y = (-1/2)x - 7/2, perpendicular line: y = 2x + 3/2.
c) For 4y - 2x = 1, solving for y gives us y = (1/4)x + 1/2.
d) Solving for y in 9x - 3y = 10 yields y = 3x - 10/3.
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