Equations of All Kinds of Lines
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 \(2\ell + 2w = 50\).
- 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 $x=a$.
Let’s write equations for vertical and horizontal lines.
Take a piece of string 50 centimeters long and tie the ends together to be used as demonstration in the third activity.
- 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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