# 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 Materials

### Required Preparation

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

### Learning Targets

### 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.

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