Lines and Linear Relationships

This course begins with a review of lines.  Why should we care about lines?  For one thing, linear relationships are all over the place.  They appear in just about every field of study that puts numbers to things.  For example,


Of course, [this relationship isn’t exactly linear it’s stochastic and that’s ok, we can do regression]


 Another reason we care about lines is that they’re among the simplest of mathematical models.  The key idea of calculus is that we can approximate difficult, nonlinear things using simpler, linear things.


Key Features of Linear Relationships

Let’s look at the line y = ½ x + 3.  For this line:

  • Draw a graph. How do the numbers ½ and 3 correspond to features in your graph?
  • Create a table of values.  What patterns can you find in the table of values?
  • Find ways to articulate connections between the graph, the equation, and the table.  Explain your connections to someone who’s willing to listen.  For example: a friend; a tutor; Marc Boucher; or me.


Let’s look at the line drawn below:


For this line:

  • Create a table of values.
  • Create an equation.  How do features in the graph correspond to numbers in your equation?
  • Look for the same connections you found above. Are they still there? In what ways are they the same? In what ways are they different?

Find the point (x,y) at which these two lines meet.  For this problem, don’t use any technology – just algebra. What are some ways you can make sense of your answer?

Moving On

Make sure that you can solve linear equations like ¾ x + 3 = 2x + 4 and that you understand these connections:


Linear relationships are so ubiquitous that this will be worth your time. It’s also the foundation on which we’re going to build things.

When you’re ready, let’s move on to quadratics.