Coordinate Geometry (CH3)

Coordinate Geometry (CH3)

1. Solving Linear Equations

   • Remember, to solve any equation we need to isolate the pronumeral we wish to find; that is, we must undo all the operations that have been performed on the pronumeral.

2. Sketching Linear Graphs

   • Written as y = mx + c
   • The gradient (or slope) is a measure of the steepness of a graph.
   • If the gradient, m, of the line is +, the graph will have an upward slope to the right. If the gradient, m, of the line is -, the graph will have a downward slope to the right.
   • x & y-intercept method
     • Value of c = y-intercept
     • This method involves finding the x- and y-intercepts, then joining them to sketch the straight line
   • gradient-intercept method
     • This method is often used if the equation is in the form y = mx + c, where m represents the gradient (slope) of the straight line, and c represents the y-intercept. Step 1: Plot a point at the y-intercept. Step 2: Write the gradient in the form m = rise run. Step 3: Starting from the y-intercept, move up the number of units suggested by the rise (move down if the gradient is negative). Step 4: Move to the right the number of units suggested by the run and plot the second point. Step 5: Rule a straight line through the two points.
   • y=mx
     • These graphs have the value of c = 0. They pass through the origin (0, 0), providing only one point to plot. This means that a second point must be determined by either: choosing any x-value, then calculating the corresponding y-value or using the gradient–intercept method.
   • y=c/x=a
     • These equations can be written in the form y = c or x = a, where c and a are both constants. Lines of the form y = c are parallel to the x-axis, having a gradient of zero and a y-intercept of c. Lines of the form x = a are parallel to the y-axis, having an undefined (infinite) gradient and no y-intercept (since they do not cross the y-axis).

3. Determining Linear Equations

   • The gradient of a straight line can be calculated from the coordinates of two points (x1, y1) and (x2, y2) which lie on the line.
     • Once the gradient has been found, substitute one pair of known x- and y-values into the standard linear equation y = mx + c to determine the value of c. If the y-intercept has already been given, then this is the value of c and no further calculation is required.
   • Point-Gradient Method
     • Sometimes the gradient and another point which is not the y-intercept is given. The value of c can then be found by substituting the coordinates of this point into y = mx + c

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