# The Equation of a Straight Line (y = mx + c)

## The Equation of a Straight Line (y = mx + c)

# The Gradient: Understanding and Implementing

- The
**gradient**of a line is effectively its ‘steepness’. - The
**gradient**(m) can be calculated using the formula:**m = (y2 - y1) / (x2 - x1)**, where (x1, y1) and (x2, y2) are the coordinates of two points on the line. - The idea of gradient can be pictorially represented as ‘rise over run’.
- The
**negative gradient**signifies that the line slopes downwards as it moves from the left to right. - A line that goes straight across (horizontal) has a gradient of zero.

# Parallel and Perpendicular Lines: Gradients and Relationships

- Lines that are
**parallel**share the same gradient. - Lines that are
**perpendicular**have slopes that multiply together to give -1. So if Line 1 has gradient m1 and Line 2 has gradient m2, for the lines to be perpendicular,**m1 * m2 = -1**.

# The Pythagorean Method: Distances Between Two Points

- The distance between two points can be calculated using
**Pythagoras’ theorem**:**√[(x_2-x_1)² + (y_2-y_1)²]**. - This forms the hypotenuse of a right triangle, where the horizontal and vertical distances between the two points are the other two sides of the triangle.

# Ratios and Coordinate Determination: Finding Points on a Line

- Any point lying between two given points (x1, y1) and (x2, y2) on a line can be found by using a
**ratio formula**: (x, y) = ((1-k)x1+ kx2 , (1-k)y1+ ky2). - By adjusting the value of k (which is the ratio), we can adjust where the point is on the line.

# Line Equations: The Slope-Intercept Form

- The equation of a straight line can be written in the form
**y = mx + c**, where m is the gradient and c is the y-intercept. - The
**y-intercept**is the point where the line intersects (crosses) the y-axis. - The
**gradient**can be found from the slope-intercept form of the equation by looking at the coefficient in front of the x variable.