Q1. What is the equation of the line passing through the points (1, 2) and (3, 4)?
Solution:
The slope m = (4 - 2) / (3 - 1) = 1. Using point-slope form, y - 2 = 1(x - 1) gives y = x + 1.
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Q2. If a line has the equation 7x - 3y + 21 = 0, what is its slope?
Solution:
Rearranging gives y = (7/3)x + 7. The slope is -7/3.
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Q3. What is the slope of the line perpendicular to the line 7x - 2y + 3 = 0?
Solution:
The slope of the line is 7/2. The slope of the perpendicular line is the negative reciprocal, which is -2/7.
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Q4. Determine the angle between the lines y = 2x + 3 and y = -1/2x + 1.
Solution:
The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan^(-1) |(m1 - m2)/(1 + m1*m2)| = tan^(-1)(5/4) which is approximately 60 degrees.
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Q5. Find the equation of the line parallel to y = 5x - 2 that passes through the point (2, 3).
Solution:
Since the slope is the same (5), using point-slope form: y - 3 = 5(x - 2) gives y = 5x - 7.
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Q6. Find the equation of the line that passes through the point (4, 5) and is perpendicular to the line y = 1/3x + 2.
Solution:
The slope of the given line is 1/3, so the slope of the perpendicular line is -3. Using point-slope form, we get y - 5 = -3(x - 4), which simplifies to y = -3x + 17.
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Q7. Find the equation of the line parallel to y = 5x - 3 that passes through the point (2, 1).
Solution:
Since the slope is the same (5), using point-slope form: y - 1 = 5(x - 2) gives y = 5x - 9.
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Q8. Find the x-intercept of the line 5x - 2y + 10 = 0.
Solution:
Setting y = 0 in the equation gives 5x + 10 = 0, thus x = -2. The x-intercept is -2.
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Q9. Find the equation of the line that passes through the point (2, -3) and has a slope of 4.
Solution:
Using point-slope form, y + 3 = 4(x - 2) simplifies to y = 4x - 11.
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Q10. Find the equation of the line that passes through (2, 3) and is perpendicular to the line y = 1/3x + 2.
Solution:
The slope of the given line is 1/3, so the perpendicular slope is -3. Using point-slope form, y - 3 = -3(x - 2) gives y = -3x + 9.
