Analytic Geometry

One of the most important aspects of analytic geometry is the idea that an algebraic equation can relate to a geometric figure. Consider the equation 2x + 3y = 44. The solution to this equation is an ordered pair (x,y) which represents a point. If the set of every point that makes the equation true (called the locus) were plotted, the resulting graph would be a straight line. For this reason, equations such as these are known as linear equations. The standard form of a linear equation is Ax + By = C, where A, B, and C are constants and A and B are not both 0. It is interesting to note that an equation such as x = 4 is a linear equation. The graph of this equation is made up of the set of all ordered pairs in which x = 4.

The steepness of a line relative to the x-axis can be determined by using the concept of the slope. The slope of a line is defined by the equation

The value of the slope can be used to describe a line geometrically. If the slope is positive, the line is said to be rising. For a negative slope, the line is falling. A slope of zero is a horizontal line and an undefined slope is a vertical line. If two lines have the same slope, then these lines are parallel.

The slope gives us another common form for a linear equation. The slope-intercept form of a linear equation is written y = mx + b, where m is the slope of the line and b is the y intercept. The y intercept is defined as the value for y when x is zero and represents a point on the line that intersects the y axis. Similarly, the x intercept represents a point where the line crosses the x axis and is equal to the value of x when y is zero. Yet another form of a linear equation is the point-slope form, y — y₁= m(x — x₁). This form is useful because it allows us to determine the equation for a line if we know the slope and the coordinates of a point.