Lines

[IloveManUtd]

In my question:

Given that the x-intercept of a line is twice its y-intercept and that the line passes through the point of the intersection of the lines 3y+x=3 and 4y+3x=5, find the equation of this line.

I do not know whether the line makes a 90-degree angle with one of the lines. And if it does, which gradient should I use. Please help.

[BigGlenntheHeavy]

[mmm4444bot]

Don't assume that the unknown line is perpendicular to either of the given two. (There's no suggestion for that.)

What coordinates do you get for the intersection point of the two given lines?

The unknown line has an equation of the form y = m*x + b.

The y-intercept is b.

The x-intercept is -b/m.

Since the latter is twice the former, we have 2b = -b/m.

Solve this equation for m.

Then, substitute the three known values into y = m*x + b, and solve for b.

You're done.

[Wally]

To find the equation of a line, you need either two points that are on the line, or you need one point and the gradient (slope). In this problem, you will use one point and the gradient. You can get the point by solving the two equations (3y+x=3 and 4y+3x=5) in order to find what point is on both lines. You can get the gradient from the fact that the x-intercept of the line is twice its y-intercept. Draw a picture to help with this.

Now that you have the equation of the line, you want to know if the line is perpendicular with either line 3y+x=3 or line 4y+3x=5. The trick to know is the following. If two lines are parallel, then they have the same gradient. If two lines are perpendicular, then there gradients are negative reciprocals of each other. For example, if the gradient of a line is 3, then a line perpendicular to that line has a gradient of - 1/3. Draw another picture of this to see why.

[mmm4444bot]

you want to know if the line is perpendicular with either line 3y+x=3 or line 4y+3x=5. Why is that?

[mmm4444bot]

Hey, Manchester United fan, I want to change my question to you.

Do you know how to find the coordinates of the intersection point of the two given lines? This is an easy first step, if you know how to work with ratios.