Special lines and the equation of a line

The equation x = 3

Here is an equation with only $x$: $x = 3$. Can it be an equation in two variables? Yes. We write it as $x + 0 \times y = 3$. For this equation, $x$ must be 3, and $y$ can be any number.

So the solutions are $(3, 0)$, $(3, 1)$, $(3, -2)$, $(3, 5)$ and so on. All of these points have $x = 3$. They lie on a vertical line, parallel to the y-axis, through the point 3 on the x-axis.

In the same way, $y = -2$ is a line that is parallel to the x-axis. Every point on it has $y = -2$.

Lines through the origin

Look at the equation $y = 2x$. If $x = 0$, then $y = 0$. So the point $(0, 0)$, the origin, is on the line. Some other solutions are $(1, 2)$, $(2, 4)$ and $(3, 6)$.

An equation $ax + by = 0$ has no constant term. It always has the solution $(0, 0)$. So its graph is a line that passes through the origin.

How steep is the line?

Look at the line $y = 2x$. When $x$ grows by 1, $y$ grows by 2. This number is the slope of the line. It tells how steep the line is.

For a line $y = mx + c$,

• $m$ is the slope. For every 1 that we go to the right, the line goes up by $m$.
• $c$ is the value of $y$ where the line meets the y-axis.

The line $y = 2x + 1$ starts at 1 on the y-axis. It goes up by 2 for every 1 to the right.

Set $m = 0$. The line lies flat, like $y = 4$. A bigger $m$ makes a steeper line, and a negative $m$ makes the line go down to the right.

A line that tells a story: the taxi fare

A taxi charges a fixed 2 dollars. Then it charges 3 dollars for every kilometre. For a journey of $x$ km, the fare $y$ is

Here the slope, 3 dollars for each kilometre, is the rate. The number 2 is the fixed charge. It is the value of $y$ at $x = 0$.

• A journey of 5 km costs $3 \times 5 + 2 = 17$ dollars.
• A fare of 26 dollars is for a journey where $3x + 2 = 26$. We want $x$, so $3x = 24$ and $x = 8$ km.

On the graph, we read a distance on the x-axis and go up to the line to find the fare.

Practise with new numbers

Try it in Python

This program finds the fare of a taxi for a few distances. It also finds the distance for a fare of 26 dollars.

Remember these

• $x = a$ is a vertical line, parallel to the y-axis. $y = b$ is a horizontal line, parallel to the x-axis.
• An equation $ax + by = 0$ gives a line through the origin.
• For $y = mx + c$, the slope is $m$, and the line meets the y-axis at $c$.
• A linear equation can describe a real situation, like a fare, with a rate and a fixed charge.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Describe the line $y = 4$.

2. Does the line $3x - y = 0$ pass through the origin?

3. A phone plan has a fixed charge of $5$ dollars and $2$ dollars for every gigabyte. Write the cost $y$ for $x$ gigabytes. What do $6$ gigabytes cost?

4. What is the slope of $y = -2x + 5$, and where does the line meet the y-axis?