Equations of Straight Lines
Master slopes, intercepts, and inclinations. Learn to build straight-line equations from any clues, and solve line intersections instantly using simultaneous equations.
1. The Core Concept
The DNA of a Line: A straight-line equation represents a geometric path on a coordinate plane. Every line is uniquely defined by two attributes: how steep it is (the slope, m) and where it crosses the axes (the intercepts). In the DSE, you must be comfortable switching between different equation forms depending on the clues given.
2. DSE Equations of Straight Lines Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Slope (Gradient) | m = (y₂ − y₁) / (x₂ − x₁) = tan θ | • m > 0: Uphill (0° < θ < 90°) • m < 0: Downhill (90° < θ < 180°) • m = 0: Horizontal line Parallel: m₁ = m₂Perpendicular: m₁ × m₂ = −1 |
| 2. Slope-Intercept Form | y = mx + c• m = slope • c = y-intercept (where x = 0) | Use this form to instantly sketch a line. The line passes through (0, c) with steepness m. |
| 3. Point-Slope Form | y − y₁ = m(x − x₁)Passes through (x₁, y₁) with slope m. | Your Go-To Formula: If given two points, find m first, then plug either point into this formula. |
| 4. General Form | Ax + By + C = 0(A, B, C ∈ ℤ and A > 0) | Fast Coefficients Read (Paper 2): • Slope m = −A/B• y-intercept = −C/B• x-intercept = −C/A |
| 5. Line Intersections | Solve L₁: y = m₁x + c₁ and L₂: y = m₂x + c₂ simultaneously.• 1 solution: Intersecting ( m₁ ≠ m₂)• 0 solutions: Parallel ( m₁ = m₂, c₁ ≠ c₂)• ∞ solutions: Coincident ( m₁ = m₂, c₁ = c₂) | Use substitution or elimination. Plug solved x back into either equation to find y. |
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💡 Exam Traps & Secrets
⚠️ The Negative Inclination Angle TrapWhen finding the angle of inclination θ for a line with a negative slope, your calculator returns a negative angle (e.g.,
m = −1 ⟹ θ = tan⁻¹(−1) = −45°). Rule: The inclination must be measured from the positive x-axis (0° ≤ θ < 180°). You must add 180° to get the correct obtuse angle: θ = −45° + 180° = 135°.⚠️ The Perpendicular Slope TrickIf given a line
3x + 4y − 12 = 0 (slope m₁ = −3/4), any line perpendicular to it must have the negative reciprocal slope: m₂ = −1/m₁ = +4/3. Use this to write perpendicular slopes instantly without working steps.⚠️ The "Undefined Slope" Trap (Vertical Lines)Horizontal lines have slope
m = 0 (equation: y = k). Vertical lines have an undefined slope (equation: x = h). Never write m = ∞ in your Paper 1 steps; simply state: "Since the line is vertical, its equation is x = h."⚠️ Point VerificationTo check if a point P
(x₁, y₁) lies on a line Ax + By + C = 0, substitute x₁ and y₁ into the LHS. If the resulting value is exactly 0, the point lies on the line. If it is not 0, the point does not lie on the line.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Two-point slope + Point-Slope conversion)Find the equation of the line passing through A
(1, 2) and B(4, 8) in General Form.(Setting y=0 and x=0 to find intercepts)Find the x-intercept and y-intercept of the line
2x + 5y − 10 = 0.(Simultaneous equations)Find the point of intersection of the lines
y = 2x + 1 and x + y = 4.(Δ < 0)Find the range of k such that the line
y = x + k does not cut the circle x² + y² − 4x = 0.