1
Find slope m = (7-3)/(4-2) = 4/2 = 2
2
Use point-slope form with point (2,3)
5
y = 2x - 1
Answer: y = 2x - 1 or 2x - y - 1 = 0
Worked Example 2: Find distance between parallel lines 3x + 4y + 5 = 0 and 3x + 4y - 7 = 0.
1
Both lines have form 3x + 4y + c = 0, so they are parallel
2
Use distance formula: d = |c1 - c2|/√(a² + b²)
3
Here c1 = 5, c2 = -7, a = 3, b = 4
4
d = |5 - (-7)|/√(3² + 4²) = 12/√25 = 12/5
Answer: 12/5 units
Shortcut 3 - Equation from intercepts: If line cuts x-axis at (a,0) and y-axis at (0,b), equation is x/a + y/b = 1.
Common Mistake - The #1 trap: Students forget that for perpendicular lines, the product of slopes is -1, NOT the sum. Many write m1 + m2 = -1 instead of m1 × m2 = -1. This single mistake costs marks in every exam.
Another frequent error is sign mistakes while applying section formula. Always check if the ratio is internal (same signs) or external (opposite signs).
Remember that when finding perpendicular distance from point to line ax + by + c = 0, the formula is |ax1 + by1 + c|/√(a² + b²). The modulus sign is crucial - distance cannot be negative.
For intersection of two lines, solve the equations simultaneously. If you get 0 = 0, lines are identical. If you get contradiction like 0 = 5, lines are parallel.