Plotting an equation as a graph shows the way that the value of one variable changes as the other changes. This uses the idea that any equation relating two real variables can be pictured as a relationship between two-dimensional Cartesian coordinates, x and y. An equation can therefore be interpreted as a curve representing the corresponding values of x and y determined by that equation.
The equation y = x2 generates a parabolic curve of points, as shown. More complicated equations can create more complicated curves, though for each x there may be none or many corresponding values of y.
When a pair of simultaneous equations are plotted on the same axes, the intersections mark points where x and y satisfy both equations. Thus, the solution of simultaneous equations is essentially a question of determining the intersection points of curves: algebra and geometry meet.
Any straight line in the plane can be written as either x = a, where a is constant (this is the special case of a vertical line) or the more standard form y = mx + c, where m and c are constants. The constant m represents the slope of the line and c is the value of y where the line meets the y-axis.
The slope, or gradient, of the line is calculated by considering any two points on the line. It is equal to the change in height between the points, divided by the change in horizontal position between the points.