# Finding Equations of Lines

Want some practice with related concepts first?

- Introduction to Equations and Inequalities in Two Variables
- Introduction to the Slope of a Line
- Practice with Slope
- Graphing Lines

Every non-vertical line in the coordinate plane can be described by an equation of the form $\,y = mx + b\,,$ where:

- $\,m\,$ is the slope of the line
- $\,b\,$ is where the line crosses the $\,y$-axis

The equation $\,y = mx + b\,$
is called the *slope-intercept form of the line*.

*Two different points* uniquely
determine a line.
*One point and a slope*
also uniquely determine a line.
This web exercise gives you practice
writing the equation of the line in these two situations.

## Example (Known Point, Known Slope)

$y = mx + b$ | A line with slope $\,3\,$ isn't vertical, so it can be described by an equation of this form. |

$y = 3x + b$ | Substitute the known slope, $\,3\,,$ in for $\,m\,.$ Next, we must find $\,b\,.$ |

$5 = 3(-1) + b$ | Since $\,(-1,5)\,$ lies on the line, substitution of $\,-1\,$ for $\,x\,$ and $\,5\,$ for $\,y\,$ makes the equation true. |

$5 = -3 + b$ | simplify |

$\,b = 8\,$ | add $\,3\,$ to both sides; write in the conventional way |

$y = 3x + 8$ | substitute the now-known value of $\,b\,$ into the equation |

Thus, the line with slope $\,3\,$ that passes through $\,(-1,5)\,$ is described by the equation $\,y = 3x + 8\,.$

Make sure you understand what this means!
Let $\,\ell\,$ denote the line
with slope $\,3\,$ that passes through the
point $\,(-1,5)\,.$
Every point that *lies on* $\,\ell\,$
has coordinates that make the
equation $\,y = 3x + 8\,$ *true*.
Every point that
*doesn't lie on* $\,\ell\,$ has coordinates
that make the equation $\,y = 3x + 8\,$ *false*.

Head up to wolframalpha.com and type in:

y = 3x + 8, x = -1, y = 5

(Cut-and-paste, if you want.) You'll see a graph of the line, with the given point indicated by crosshairs. By adding in an additional set of crosshairs, you can see that going up $\,3\,$ and to the right $\,1\,$ brings you to another point on the line:

y = 3x + 8, x = -1, y = 5, x = 0, y = 8

## Example (Two Known Points)

$y = mx + b$ | start with slope-intercept form |

$y = -3x + b$ | substitute the now-known slope, $\,-3\,,$ in for $\,m\,$ |

$4 = -3(-1) + b$ | Which point should you use? It doesn't matter! In general, try to choose the simplest numbers to work with. |

$4 = 3 + b$ | simplify |

$\,b = 1\,$ | subtract $\,3\,$ from both sides; write in the conventional way |

$y = -3x + 1$ | substitute the now-known value of $\,b\,$ into the equation |

You might want to check that the two points do indeed lie on the line:

$-5\ \overset{\text{?}}{ = } -3(2) + 1\,$
Check!

$4\ \overset{\text{?}}{ = } -3(-1) + 1\,$
Check!