# A line passing through the origin has slope −1. Determine the value of a?

A line passing through the origin has slope −1. Determine the value of a, if the line is to contain the point (a, 4)

Relevance

We can approach this from two direction, the purely logical way or the intuitive way and I'll do both.

From intuition we see that the gradient is -1 so it's at an angle of 45 degrees from the x axis and we're given the points (a,4). Due to the line being so nice and even we can easily assume just from imagining the line, that a = -4 since the ratio between x and y is -1, so whatever x does y does the same.

From logic what we do is use the equation for a straight line.

y-b=m(x-a) Where "a" is an x coordinate and "b" is a y coordinate and m is the gradient.

So let's substitute what we know into this:

y-4=-1(x-a)

y=-x+a+4

So how do we find out the constant a? What we do is we set y and x to 0 in order to get them out of the equation.

So 0=a+4 so a=-4.

Hope that helps :)

• The equation of a line with slope −1 is given as

y=−x +K

Passes through origin

→0=−0 +K Hence K= 0

and equation is y=−x . It contains (a, 4) hence

4=−a or a = −4

• The "foundation" is yet another call for the element (0,0). yet "the muse has slope 2" would not make any sense. even nevertheless the different sentence ("A line passing interior the path of the muse has slope 2.") does make sense. the final equation for a line is y = mx + b, the place m is the slope and b is the y-intercept. because of the fact the slope is two, which means y = 2x + b. and because (0,0) is a element on the line, then 0 = 2(0) + b, so b=0. this means that the equation for the line is y = 2x. If this line additionally includes (a, -a million), then -a million = 2(a) fixing this for a supplies a = -a million/2.

• Any line passing through the origin will be of the form y=x.(reason is that the general equation is y=mx+c.As the line passes through origin the y intercept(c) will be zero resulting in y=mx). But this is for a line with slope 1. So the required line will be y=-x(this line passes through origin and has slope -1). Substituting the given point we get 4=-a. this means a=-4.

• at the origin --> (0, 0)

m = -1

y = mx + b

y = -1(x) + 0

y = -x

now,given (a, 4)

y = -x

y = -a

4 = -a

a = -4 . ..