Lesson 11

Writing Equations for Lines

Lesson Narrative

The previous lesson introduces the idea of slope for a line. In this lesson, the slope is used to write a relationship satisfied by any point on a line. The key idea is to introduce a general or variable point on a line, that is a point with coordinates \((x,y)\). These variables \(x\) and \(y\) can take any values as long as those values represent a point on the line. Because all slope triangles lead to the same value of slope, this general point can be used to write a relationship satisfied by all points on the line. 

Slope triangle with unknowns.

In this example, the slope of the line is \(\frac{1}{3}\) since the points \((1,1)\) and \((4,2)\) are on the line. The slope triangle in the picture has vertical length \(y-1\) and horizontal length \(x-1\) so this gives the equation \(\displaystyle \frac{y-1}{x-1} = \frac{1}{3}\) satisfied by any point on the line (other than \((1,1)\)). This concise way of expressing which points lie on a line will be developed further in future units.

Learning Goals

Teacher Facing

  • Create an equation relating the quotient of the vertical and horizontal side lengths of a slope triangle to the slope of a line.
  • Justify (orally) whether a point is on a line by finding quotients of horizontal and vertical distances.

Student Facing

Let’s explore the relationship between points on a line and the slope of the line.

Learning Targets

Student Facing

  • I can decide whether a point is on a line by finding quotients of horizontal and vertical distances.

CCSS Standards

Building On


Glossary Entries

  • similar

    Two figures are similar if one can fit exactly over the other after rigid transformations and dilations.

    In this figure, triangle \(ABC\) is similar to triangle \(DEF\).

    If \(ABC\) is rotated around point \(B\) and then dilated with center point \(O\), then it will fit exactly over \(DEF\). This means that they are similar.

  • slope

    The slope of a line is a number we can calculate using any two points on the line. To find the slope, divide the vertical distance between the points by the horizontal distance.

    The slope of this line is 2 divided by 3 or \(\frac23\).

    A graph in the coordinate plane with a line on it.

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