# 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.

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.

### Student Facing

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

Building On