Lines, Slopes & Cuts Discover the story every straight line tells.

The Coordinate Grid

Coordinate Plane

Two perpendicular number lines—the horizontal x-axis and vertical y-axis—meet at the origin \( (0,0) \). Locations are written as ordered pairs \( (x,y) \).

Key Characteristics:

  • Horizontal x-axis shows left–right values.
  • Vertical y-axis shows up–down values.
  • Axes intersect at the origin \( (0,0) \).
  • Each point is written as \( (x,y) \).

Example:

Plot \( (-2,3) \) — it sits in Quadrant II.

What Is Gradient?

Term

Gradient (Slope)

Definition

Measure of a line’s steepness — how far it rises or falls for each unit you move right.

  • Rise ÷ Run
  • Positive: Upward tilt
  • Negative: Downward tilt

Example: Gradient 2 means the line rises 2 units for every 1 unit across.

Seeing Gradient

rise-run gradient visual

Arrows show 6-unit rise and 4-unit run.

Gradient = Rise ÷ Run

Vertical rise = 6 units; horizontal run = 4 units.

Therefore \( \text{gradient} = \frac{6}{4} = 1.5 \).

Key Points:

  • rise = change in \(y\)
  • run = change in \(x\)
  • gradient shows line steepness

Gradient Formula

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

The slope formula uses any two distinct points on a non-vertical straight line; the ratio is constant.

Variable Definitions

\(m\) gradient (slope)
\(x_1\) x-coordinate of first point
\(y_1\) y-coordinate of first point
\(x_2\) x-coordinate of second point
\(y_2\) y-coordinate of second point

Applications

Find slope

Calculate how steep a line is between two given points.

Write line equation

Use \(m\) with one point to form \(y = mx + c\).

Match Rise & Run

Active practice: drag each run number to the corresponding rise so the gradient \(m=\frac{\text{rise}}{\text{run}}\) is correct.

Draggable Items

4
2
1
5

Drop Zones

Rise 2, m = 1/2

Rise 4, m = 2

Rise -3, m = -3

Rise 5, m = 1

Tip:

Remember: \(m=\frac{\text{rise}}{\text{run}}\). Swap rise or run to get the needed ratio.

Meet the Intercepts

x-Intercept

Point where the line cuts the x-axis.
Lies on \(y = 0\).
Written as \((a, 0)\).

y-Intercept

Point where the line cuts the y-axis.
Lies on \(x = 0\).
Written as \((0, b)\).

Key Similarities

Both are points where the line meets an axis.
A straight line has one of each.
Help sketch the graph quickly.
Values found by setting the other variable to zero.

y = mx + c

\[y = mx + c\]

Variable Definitions

\(m\) gradient (slope)
\(c\) y-intercept \( (0,c) \)
\(x\) horizontal coordinate
\(y\) vertical coordinate

Applications

Quick Sketch

Plot \(c\) on the y-axis, move with slope \(m\), draw the straight line.

Model Checking

Identify slope and intercept to test if data fits a straight-line equation.

Full Picture

https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/sDn7keXqkYFAjx0mCvRGuBlMIKmeE8VRaAcPN8Sc.png

Line \(y = -0.5x + 3\)

Read the equation to pick out each intercept and the slope.

These values let you quickly place the line on the graph.

Key Points:

  • Gradient \(m = -0.5\)  (negative)
  • y-intercept: set \(x = 0\) → point \((0, 3)\)
  • x-intercept: set \(y = 0\) → solve \(0 = -0.5x + 3\) gives \(x = 6\); point \((6, 0)\)

Multiple Choice Question

Question

What is the gradient of the line through \( (2,1) \) and \( (6,5) \)?

1
1
2
2
3
4
4
\( \frac{1}{2} \)

Hint:

Use \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Key Takeaways

Gradient (slope) equals rise / run and shows a line’s steepness.

The y-intercept is the point where the line meets the y-axis \((x = 0)\).

The x-intercept is where the line crosses the x-axis \((y = 0)\).

Equation \(y = mx + c\) links gradient \(m\) and y-intercept \(c\) in one formula.

Knowing \(m\) and the intercepts lets you sketch or interpret any straight line quickly.

Thank You!

We hope you found this lesson informative and engaging.