Straight Lines

Straight Lines – Study Notes

(Manipal MET – Mathematics)

1. What Is a Straight Line?

• A straight line is a set of points that keeps going forever in two opposite directions.
• Think of a road that never bends.

2. The Two‑Point Way

• If you know two points on a line, you can draw it.
• Example: points A(1, 2) and B(3, 6).

3. Slope (How Steep?)

• Slope tells how steep a line is.

• Analogy: the steeper the hill you walk, the more you climb per meter you move forward.

• Formula (no jargon):

  slope = (change in height) ÷ (change in distance)
         = (y₂ – y₁) ÷ (x₂ – x₁)
  

• Example: between (1, 2) and (3, 6)

  slope = (6 – 2) ÷ (3 – 1) = 4 ÷ 2 = 2
  

  The line rises 2 units for every 1 unit to the right.

4. The Equation of a Line

4.1 Slope‑Intercept Form

y = mx + b

• m is the slope.

• b is where the line cuts the vertical axis (the y‑axis).

• Example: Using slope 2 and point (1, 2)

  2 = 2·1 + b    →   b = 0
  y = 2x
  

  The line goes through the origin (0, 0).

4.2 Point‑Slope Form

y – y₁ = m (x – x₁)

• Start from a known point (x₁, y₁).

• Plug the slope you found.

• Example: point (1, 2), slope 2

  y – 2 = 2(x – 1)
  

4.3 Standard Form (Optional)

Ax + By = C

• A, B, C are whole numbers.
• Not needed for beginners but useful later.

5. Parallel and Perpendicular Lines

• Analogy: two roads going in the same direction stay parallel; a road that turns into a cross‑road is perpendicular.

6. Graphing a Line

1. Plot the intercepts
   • y‑intercept (b) is the point where the line meets the y‑axis.
   • x‑intercept (set y = 0 and solve for x).
2. Use the slope
   • From a point, go up/down m units and right/left 1 unit to land on another point.
3. Draw the line
   • Connect the points and extend both ways.

Tiny Example:
Line: y = 2x + 1

• y‑intercept = 1 → point (0, 1).
• Slope = 2 → from (0, 1) go right 1, up 2 → point (1, 3).
• Draw the line through (0, 1) and (1, 3).

7. Quick Checklist

• Find slope before writing the equation.
• Use point‑slope form if you have a point.
• Use slope‑intercept to see the y‑intercept instantly.
• Check parallel/perpendicular by comparing slopes.

8. Practice (Optional)

1. What is the slope of a line that passes through (0, 4) and (2, 0)?
2. Write the equation of a line with slope –3 that passes through (5, 1).
3. Two lines: y = 4x + 2 and y = 4x – 3. Are they parallel?
4. Find a line perpendicular to y = –½x + 6.

(Try solving them before checking the answers below.)

Answers

1. Slope = (0 – 4) ÷ (2 – 0) = –4 ÷ 2 = –2.
2. Use point‑slope: y – 1 = –3(x – 5) → y = –3x + 16.
3. Yes, both have slope 4 → parallel.
4. Perpendicular slope = reciprocal of 4 → –¼ → line: y = –¼x + b (use a point to find b if needed).

Final Thought

A straight line is like a straight road: it keeps going, it has a slope (steepness), and it can be described with a simple equation. Once you know how to read the slope and intercept, you can draw and work with any line you meet in math. Happy exploring!