The General Equation of a straight line There
- Slides: 8
The General Equation of a straight line There are 2 main ways of writing the equation of a straight line: The gradient/intercept form y = mx + c m = gradient c = y-intercept General form ax + by + c = 0 a>0 a, b, and c are * whole numbers * “reduced” to lowest integers
Example 1 Write y = 3 x + ½ in general form. y = 3 x + ½ 2 y = 6 x + 1 6 x – 2 y + 1 = 0 Example 2 What is the gradient and y-intercept of the line 8 x + 4 y – 12 = 0? Is the line 8 x + 4 y – 12 = 0 in general form? 8 x + 4 y – 12 = 0 4 y = – 8 x + 12 y = – 2 x + 3 m = – 2 and c = 3
Gradient between two points The gradient of a line is a measurement of its steepness. The gradient is the ratio of the rise to the run of the line. We use the letter m for gradient. y (x 2, y 2) (y 2 y 1) For any line. Pick 2 points (x 1, y 1) and (x 2, y 2). (x 1, y 1) Draw a right angled triangle Then use “change in y” over “change in x” (x 2 x 1) x
Gradient between two points The gradient of a line may be small, having a “gentle” slope The gradient of a line may be large, having a “steep” slope A line with a positive gradient slopes “upwards” from left to right. A line with a negative gradient slopes “downwards” from left to right.
Example 2 Show the points A(2, 5), B(6, 11) and C(– 6, – 7) lie on a straight line Mathematical word for this is “collinear” First calculate the gradient of any 2 points, say AB. Then calculate the gradient of either AC or BC What would happen if we had the points the other way around? . As m. AB = m. AC A, B and C are collinear.
The point gradient formula The equation of a straight line through a given point with a given gradient can be found using: y – y 1 = m(x – x 1) m = gradient (x 1, y 1) is a point on the line.
Example 1 What is the equation of the line through the point (2, 7) with a gradient of ¾? y – y 1 = m(x – x 1) y – 7 = ¾(x – 2) 4 y – 28 = 3 x – 6
Example 2 What is the equation of the line through the point (3, – 4) with a gradient of – 5? y – y 1 = m(x – x 1) y – (– 4) = – 5(x – 3) y + 4 = – 5 x + 15 y = – 5 x + 11