Introduction
The line and its cartesian equation are related in a very special way. Think of the line as an infinite collection of points, all of which lie in a straight line. Each point is uniquely addressed by cartesian coordinates.
The critical thing to remember is that the coordinates of every single point that is on the line, when substituted into the lines equation, will make that equation true. In other words, points on the line will satisfy the equation. Points not on the line will not satisfy the equation.
When any two points can be found that both satisfy the same linear equation, a line can be sketched by simply placing a ruler onto these points and drawing the line. There is only one line that can be drawn through two given points.
Sketch a linear function using its intercepts
Another way to sketch a linear function is by finding the x-intercept and y-intercepts.
The y-coordinate of the x-intercept is always 0. Therefore to find the coordinates of the x-intercept, substitute y=0 into the equation and solve for x.
Similarly, the y-coordinate of the y-intercept is always 0. To find the coordinates of the y-intercept, simply substitute x=0 into the equation and solve for y.
In cases where the x- and y-intercepts are equal (usually at the origin (0,0)) choose another point, say x=1, and substitute x=1 into the equation and solve for y to find a second point that lies on that line. Then join the lines by using a rule to draw a straight line between them.
Equations of lines come in all forms.
The general form is usually written as Ax+By=C but can be written with the constant term (C) on the other side of the equation.
The point/slope form y=a+bx where the slope of the line is given by the coefficient b and the y-intercept given by a.
In all forms, we can use the strategy of setting x=0 and then y=0 to reveal the two intercepts.
Examples
Example 1
Consider the linear equation -6x+2y-12=0.
Find the y-value at the point where x=0.
Find the x-value at the point where y=0.
Use these points to graph the line.
The x-intercept occurs at the point where y=0.
The y-intercept occurs at the point where x=0.
We can use the points of x and y-intercepts to sketch a line.