   SEARCH HOME Math Central Quandaries & Queries  Question from Samaira, a student: Given Y=2/3 + 3/4x Give the equation that is parallel to the given one and which touches the x-axis in x=3 We have two responses for you

Hi Samaira.

Here's an example of how to solve this kind of question.
Find the equation of the line parallel to y = 10 + 4x which touches
the x-axis at x=5.

A parallel line will have the same slope as this one.

What is the slope? Remember that you can write lines in the form y=mx
+ b where m is the slope. So if we re-arrange the first line
equation, we get y = 4x + 10, so the slope is 4 because that's what is
in place of the m.

That means that the parallel line is y = 4x + ?. We need to know what
the ? is. The question also tells us this parallel line touches the
x-axis at x=5. That means it goes through the point (5, 0). So when
x = 5, then y = 0. We can substitute 5 for x and 0 for y and then
solve for ?.

y = 4x + ?
0 = 4(5) + ?
0 = 20 + ?
-20 = ?

So that means the answer is y = 4x -20.

Try it the same way with your question, Samaira.

Cheers,
Stephen La Rocque

Hi Samaira,

In order to write the equation of any line, all you need to know is the slope of the line and the (x, y) coordinates of any point that lies on the line.

Slope: given that the equation you seek must be parallel to y = 2/3 + 3/4x, you must use the same slope as the line y = 2/3 + 3/4x. Written in this fashion, the slope is the number in front of x, in other words, the slope is 3/4.

Point: you are told the line must touch the x-axis at 3, so the point (3, 0) must lie on the line. We know the y-coordinate of the point must be zero for any point on the x-axis.

Using the point-slope formula: y - y1 = m(x - x1) where m is the slope and (x1, y1) are the coordinates of the point, we substitute and simplify:

y - 0 = 3/4(x -3)
y = 3/4x - 9/4

This is the equation parallel to y = 2/3 + 3/4x and it touches the x-axis at 3.

Hope this helps,
Leeanne     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.