Forms of Linear Equations
When writing linear equations that rely on points on the line and the slopes of lines, we primarily work with two forms of linear equations: point-slope form, and slope-intercept form.
Equations written in slope-intercept form rely on the line's slope, m, and the y-coordinate of the y-intercept, b, to form the equation.
Equations written in point-slope form rely on the line's slope, m, and a point on the line (x1, y1) to form the equation.
Using the Slope and a Point on the Line
When given information about a line's slope and a point on the line, it is easiest to write the equation in point-slope form, since the pertinent information needed to form the equation is already given to us. Here is an example:
Write the equation of a line with a slope of –2 that passes through the point (4, 9).
While developing the formula in point-slope form was certainly our easiest option, often times having our equation in slope-intercept form is helpful so that we may graph the line. Slope-intercept form isolates y onto one side of the equation, with mx+b on the other side of the equation. Let's see how we can take our equation in point-slope form, and rewrite it so that we can more easily graph this line:
Using Two Points on the Line
When we are only given information about two points on the line, and no information about the slope of the line, we can use the same processes above, we just need to calculate the line's slope ourselves. To do this, we use the following formula:
where (x1, y1) and (x2, y2) are the two points on the line. In other words, we can find the difference in y-coordinates and divide it by the difference in x-coordinates to find the slope. This we can plug-in to our equation for the line. From there, we can take any one of our two points to write the equation in point-slope form. Here is an example:
Find the equation of a line that passes through the points (–2, 5) and (3, 12).
First we need to use the two points to calculate the slope:
Now that we know the slope of the line, we can use either point to write the equation in point-slope form:
Just like with our example before, we can rewrite this equation into slope-intercept form, so that we may more easily graph this line:
Using Parallel and Perpendicular Lines
Sometimes we are given information about a line, and then asked to write the equation to a line parallel or perpendicular to it. In these cases, we need to apply the relationship between the slopes of parallel lines and the slopes between perpendicular lines.
It is probably more challenging to work with problems involving perpendicular lines, so we will go through an example:
Find the equation of the line perpendicular to that passes through the point (–3, 2)