Every point on a horizontal line has the same y-coordinate. If that coordinate is c, the equation of the line is y=c and its gradient is 0. Every point on a vertical line has the same x-coordinate. If that coordinate is a, the equation of the line is x=a and its gradient is undefined. Along y=2 the y-coordinate never changes, so the vertical change between any two points is 0 and the gradient is 0. Along x=3 the horizontal change is 0. The gradient would need a division by 0, so it is undefined and x=3 cannot be written as y=mx+c. Why the equation is so short: an equation tells you which points lie on the line. On the line y=2, a point lies on it whenever its y-coordinate is 2, whatever its x-coordinate is, so no x term is needed. This is y=mx+c with m=0. Method - To find the equation from two points, compare their coordinates. The same y-coordinate c gives the horizontal line y=c; the same x-coordinate a gives the vertical line x=a. - To draw y=c, rule a horizontal line through (0,c) on the y-axis. To draw x=a, rule a vertical line through (a,0) on the x-axis. - The x-axis is the line y=0 and the y-axis is the line x=0. - The lines x=a and y=c always meet at the point (a,c). Example The points P(-4,6) and Q(3,6) both have y-coordinate 6, so the line PQ is y=6. Its gradient is 0. The line through R(2,-5) parallel to the y-axis is vertical, so it is x=2. Its gradient is undefined. The lines y=6 and x=2 meet at (2,6). Check: read the equation as a sentence about coordinates. x=2 means “every point has x-coordinate 2”, which is a vertical line, even though the equation contains the letter x. A common error is to draw x=2 as a horizontal line.

Horizontal and vertical lines

Secondary 2

Every point on a horizontal line has the same $y$-coordinate. If that coordinate is $c$, the equation of the line is $y=c$ and its gradient is $0$. Every point on a vertical line has the same $x$-coordinate. If that coordinate is $a$, the equation of the line is $x=a$ and its gradient is undefined.

Along $y=2$ the $y$-coordinate never changes, so the vertical change between any two points is $0$ and the gradient is $0$.

Along $x=3$ the horizontal change is $0$. The gradient would need a division by $0$, so it is undefined and $x=3$ cannot be written as $y=mx+c$.

Why the equation is so short: an equation tells you which points lie on the line. On the line $y=2$, a point lies on it whenever its $y$-coordinate is $2$, whatever its $x$-coordinate is, so no $x$ term is needed. This is $y=mx+c$ with $m=0$.

Method

To find the equation from two points, compare their coordinates. The same $y$-coordinate $c$ gives the horizontal line $y=c$; the same $x$-coordinate $a$ gives the vertical line $x=a$.
To draw $y=c$, rule a horizontal line through $(0,c)$ on the $y$-axis. To draw $x=a$, rule a vertical line through $(a,0)$ on the $x$-axis.
The $x$-axis is the line $y=0$ and the $y$-axis is the line $x=0$.
The lines $x=a$ and $y=c$ always meet at the point $(a,c)$.

Example

The points $P(-4,6)$ and $Q(3,6)$ both have $y$-coordinate $6$, so the line $PQ$ is $y=6$. Its gradient is $0$.

The line through $R(2,-5)$ parallel to the $y$-axis is vertical, so it is $x=2$. Its gradient is undefined.

The lines $y=6$ and $x=2$ meet at $(2,6)$.

Check: read the equation as a sentence about coordinates. $x=2$ means “every point has $x$-coordinate $2$”, which is a vertical line, even though the equation contains the letter $x$. A common error is to draw $x=2$ as a horizontal line.

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