Gradient And Equation Of Straight Line Explained for SHS 1 Core Mathematics (Semester 2, Week 1)
Straight lines appear everywhere in everyday life, from roads and roofs to graphs and construction designs.
What You Will Learn
- The meaning of gradient
- How to calculate the gradient of a line
- How to determine the equation of a straight line
- The relationship between parallel and perpendicular lines
- How to calculate distance and midpoint between points
Main Explanation
The gradient of a line tells us how steep the line is. It also shows whether the line rises or falls as it moves from left to right. A positive gradient means the line slopes upward, while a negative gradient means it slopes downward.
The gradient of a straight line is calculated using the formula:
m = ( y2 − y1 ) / ( x2 − x1 )
For example, if a line passes through the points (−4, 5) and (4, 17), the gradient is:
m = ( 17 − 5 ) / ( 4 − ( −4 ) )
m = 12 / 8 = 1.5
This means the line rises by 1.5 units for every horizontal movement of one unit.
The equation of a straight line is usually written in the form:
y = mx + c
In this equation:
- m is the gradient.
- c is the y-intercept.
To find the equation of a line, the point-gradient formula is used:
y − y1 = m ( x − x1 )
Parallel lines always have the same gradient. For example, y = 3x + 1 and y = 3x + 12 are parallel because both have gradient 3.
Perpendicular lines intersect at 90°. Their gradients are negative reciprocals of each other. If one line has gradient 2, the perpendicular line has gradient − 1 / 2.
The distance between two points can be found using the distance formula:
|PQ| = √ ( ( x2 − x1 )2 + ( y2 − y1 )2 )
The midpoint formula helps determine the point exactly halfway between two coordinates:
Midpoint = ( ( x1 + x2 ) / 2 , ( y1 + y2 ) / 2 )
Relationships Between Straight Lines
| Type Of Line | Gradient Relationship | Meaning |
|---|---|---|
| Parallel Lines | Equal gradients | Lines never meet |
| Perpendicular Lines | Negative reciprocal gradients | Lines meet at 90° |
| Positive Gradient | m > 0 | Line rises from left to right |
| Negative Gradient | m < 0 | Line falls from left to right |
Worked Examples
Example 1
Problem: Find the gradient of the line joining A(6, 0) and B(0, 3).
- Use the formula m = ( y2 − y1 ) / ( x2 − x1 ).
- Substitute the coordinates.
- m = ( 3 − 0 ) / ( 0 − 6 ).
- m = 3 / −6.
- m = − 1 / 2.
Answer: The gradient of the line is − 1 / 2.
Example 2
Scenario: A courier company charges a fixed amount plus an additional amount for every kilometre travelled.
Explanation: The graph representing the courier charges has a gradient of 3, meaning the company charges GH¢ 3 per kilometre in addition to a fixed starting charge.
Why This Topic Matters
Gradient and straight-line equations are important in engineering, architecture, transportation, economics, and science. Roads, bridges, roofs, and graphs all rely on an understanding of gradients and linear relationships. These concepts also help learners solve coordinate geometry problems and interpret graphical information.
Quick Practice
- Find the gradient of the line joining the points (3, 4) and (8, 14).
- Determine whether the lines y = 3x + 1 and y = 3x + 12 are parallel.
- Find the midpoint of the line joining the points (3, 4) and (5, 10).
Summary
The gradient measures the steepness and direction of a line. Straight-line equations are written in the form y = mx + c. Parallel lines have equal gradients, while perpendicular lines have negative reciprocal gradients. Learners can also determine distances and midpoints between points using coordinate formulas. These concepts are essential for understanding coordinate geometry and solving real-world mathematical problems.
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