Understanding Linear Relationships

• Linear relationship is a straight-line relationship between two variables.
• It is written in the form: y = mx + c
• y represents the dependent variable
• x represents the independent variable
• m represents the gradient (slope)
• c represents the y-intercept

Gradient (Slope) Formula

• The gradient measures the steepness of a line.
• m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
• If m > 0, the line slopes upwards.
• If m < 0, the line slopes downwards.
• If m = 0, the line is horizontal.
• If m is undefined, the line is vertical.

Finding the Equation of a Line

• Identify the gradient m.
• Find the y-intercept c.
• Substitute into y = mx + c.

X- and Y-Intercepts

• To find the X-Intercept: The line crosses the x-axis (y = 0). Solve y = mx + c for x.
• To find the Y-Intercept: The line crosses the y-axis (x = 0). The value of c in y = mx + c.

Special Linear Graphs

• Horizontal Line: y = c as the gradient = 0.
• Vertical Line: x = a as the gradient is undefined.

Parallel and Perpendicular Lines

• Parallel lines have the same gradient (m1 = m2).
• Perpendicular lines have negative reciprocal gradients (m1 * m2 = -1).

Plotting a Linear Graph

• Find at least two points on the line.
• Mark the y-intercept.
• Use the gradient to find another point (rise/run).
• Draw a straight line through the points.

Simultaneous Equations (Graphical Method)

• To graphically solve two equations, plot both equations on the same graph.
• The point where they intersect represents the solution (x, y).

Word Problems & Special Techniques

• Tables of Values: Substitute values of x into the equation to get y.
• Checking Solutions: Substitute points into equations to verify if they lie on the line.
• Finding Equations from Graphs: Identify gradient and y-intercept from a graph and use y = mx + c.