The line 𝐴𝐵 is parallel to the 𝑦-axis. If the coordinates of the points 𝐴 and 𝐵 are 𝑚, two and eight, six, respectively, find the value of 𝑚.
If two straight lines are parallel, then their slopes or gradients are equal. As 𝐴𝐵 is parallel to the 𝑦-axis, then the slope of 𝐴𝐵 will be the same as the slope of the 𝑦-axis. The 𝑦-axis is a vertical line. This means that the slope is undefined. Therefore, line 𝐴𝐵 will also have a slope that is undefined.
The slope or gradient of any line is calculated using the formula 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one. Substituting in the coordinates of 𝐴 and 𝐵 — 𝑚, two and eight, six — gives us that the slope is equal to six minus two divided by eight minus 𝑚. However, as the slope of 𝐴𝐵 is undefined, the denominator in this case eight minus 𝑚 must be equal to zero.
Therefore, to work out the value of 𝑚, we need to solve the equation eight minus 𝑚 equals zero. Adding 𝑚 to both sides of this equation gives us our solution 𝑚 equals eight. This means that the coordinates of 𝐴 are eight, two.
We could actually go a step further and say that any point on the line 𝐴𝐵 will have an 𝑥-coordinate of eight. This is because the equation of the vertical line that passes through eight, two and eight, six is 𝑥 equals eight.