# Lesson 11

Equations of All Kinds of Lines

### Lesson Narrative

In previous lessons, students have studied lines with positive and negative slope and have learned to write equations for them, usually in the form $$y = mx + b$$. In this lesson, students extend their previous work to include equations for horizontal and vertical lines. Horizontal lines can still be written in the form $$y = mx + b$$ but because $$m = 0$$ in this case, the equation simplifies to $$y = b$$. Students interpret this to mean that, for a horizontal line, the $$y$$ value does not change, but $$x$$ can take any value. This structure is identical for vertical lines except that now the equation has the form $$x = a$$ and it is $$x$$ that is determined while $$y$$ can take any value.

Note that the equation of a vertical line cannot be written in the form $$y = mx + b$$. It can, however, be written in the form $$Ax + By = C$$ (with $$B$$ = 0). This type of linear equation will be studied in greater detail in upcoming lessons. In this lesson, students encounter a context where this form arises naturally: if a rectangle has length $$\ell$$ and width $$w$$ and its perimeter is 50, this means that $$2\ell + 2w = 50$$

### Learning Goals

Teacher Facing

• Comprehend that for the graph of a vertical or horizontal line, one variable does not vary, while the other can take any value.
• Create multiple representations of linear relationship, including a graph, equation, and table.
• Generalize (in writing) that a set of points of the form $(x,b)$ satisfy the equation $y=b$ and that a set of points of the form $(a,y)$ satisfy the equation $x=a$.

### Student Facing

Let’s write equations for vertical and horizontal lines.

### Required Preparation

Take a piece of string 50 centimeters long and tie the ends together to be used as demonstration in the third activity.

### Student Facing

• I can write equations of lines that have a positive or a negative slope.
• I can write equations of vertical and horizontal lines.

Building On