Write linear equations using slope-intercept, point-slope, and standard form.
Given any two points on a coordinate plane, there is exactly one line that passes through these two points. Let’s start with discussing standard form since this can be used for any line (function or non-function). The standard form of a line is \(Ax+By=C\) where \(A\), \(B\), and \(C\) are constants and \(A\) and \(B\) are not both zero. If possible, you often try to write an equation in standard form where \(A\), \(B\), and \(C\) are integers and \(A\) is non-negative. Below are the different scenarios where \(A\) or \(B\) could be zero.
Given any two points on a coordinate plane, there is exactly one line that passes through these two points. Let’s start with discussing standard form since this can be used for any line (function or non-function). The standard form of a line is \(Ax+By=C\) where \(A\), \(B\), and \(C\) are constants and \(A\) and \(B\) are not both zero. If possible, you often try to write an equation in standard form where \(A\), \(B\), and \(C\) are integers and \(A\) is non-negative. Below are the different scenarios where \(A\) or \(B\) could be zero.
Neither \(A\) nor \(B\) are zero |
\(A=0\) |
\(B=0\) |
\(2x-3y=15\) |
\(0x+y=-4\) |
\(x+0y=3\) |
red line |
blue line |
green line |
As you can see, an equation such as \(x=3\) is not a function. This line has an undefined slope and can only be written using standard form.
For lines that are functions, there are two other forms that are commonly used and usually more helpful in providing important information about the line. These two forms are slope-intercept form and point-slope form. Slope-intercept form is \(y=mx+b\) where \(m\) is the slope and \(b\) is the -intercept. Point-slope form is \(y-y_1=m(x-x_1)\) where the slope is \(m\) and \((x_1,y_1)\) is any point on the line. Also remember that the equation for the slope of a line is \(\frac{y_2-y_1}{x_2-x_1}\).
Three Forms of Linear Equations
Standard Form: \(Ax+By=C\)
Slope-intercept Form: \(y=mx+b\)
Point-slope Form: \(y-y_1=m(x-x_1)\)
Standard Form: \(Ax+By=C\)
Slope-intercept Form: \(y=mx+b\)
Point-slope Form: \(y-y_1=m(x-x_1)\)
Let’s now look at how we can use these to write equations of lines.
Example 1: Write the equation of the line, in slope-intercept form, that passes through the points \((5, -6)\) and \((-1, 2)\).
Since we do not have the \(y\) -intercept yet, we will start with point-slope form.
\(\begin{align}m=\frac{2-\left(-6\right)}{-1-5}&=\frac{8}{-6}=-\frac{4}{3}\ & \ \ \ \ &\text{1) Find the slope between the two points.}\\y-2&=-\frac{4}{3}\left(x+1\right)\ & \ \ \ \ &\text{2) Use the slope and either point in point-slope form.}\\y-2&=-\frac{4}{3}x-\frac{4}{3}\ & \ \ \ \ &\text{3) Distribute} -\frac{4}{3}.\\y&=-\frac{4}{3}x+\frac{2}{3}\ & \ \ \ \ &\text{4) Add}\ 2\ \text{to both sides to solve for y}.\end{align}\)
Example 2: Write the equation of the line, in standard form, that is perpendicular to the line \(y=\frac{1}{2}x-11\) and passes through the point \((3, -7)\).
(Recall that parallel lines have the same slope and perpendicular lines have slopes that are opposite reciprocals.)
Since the slope of the given line is \(\frac{1}{2}\), the slope of a line perpendicular to that line must be \(-2\).
\(\begin{align}y-7&=-2(x-3)\ & \ \ \ \ &\text{1) Use the slope and point in point-slope form.}\\y+7&=-2+6\ & \ \ \ \ &\text{2) Distribute} -2.\\y&=-2x-1\ & \ \ \ \ &\text{3) Subtract}\ 7\ \text{from both sides. Equation is now in slope-intercept form.}\\2x+y&=-1\ & \ \ \ \ &\text{4) Add}\ 2x\ \text{to both sides to write in standard form.}\end{align}\)
Example 3: Write the equation of the line, in standard form, that is parallel to \(y=\frac{3}{5}x-10\) and passes through the point \((-4, -1)\).