Without drawing the graph, find out whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident: 9x-10y=21 & 3/2x-5/3y=7/2

The easiest way to solve this question is to write out both equations in the form y = mx + c, where m is the gradient and c is the y-intercept.

a) Thus, if we start with 9x - 10y = 21, then we get:

9x - 10y = 21

(9/10)x - y = 21/10 (Divide both sides by 10)

(9/10)x = y + 21/10 (Add y to both sides)

(9/10)x - 21/10 = y (Subtract 21/10 from both sides)

Thus, our first equation may be written as y = (9/10)x - 21/10

b) Now if we take the second equation, (3/2)x - (5/3)y = 7/2, we can follow the same process to get:

(3/2)x - (5/3)y = 7/2

(9/10)x - y = 21/10 (Multiply each side by 3/5)

(9/10)x = y + 21/10 (Add y to each side)

(9/10)x - 21/10 = y (Subtract 21/10 both sides)

Thus, the second equation may be written as y = (9/10)x - 21/10.

Now you might have already realised this but the two equations are actually exactly the same; if they are the same line then they are said to be coincident.

Note that if the two lines are parallel, then their gradients (m) would be the same, but the y-intercepts (c) would be different (eg. y = 2x + 3 and y = 2x + 4 are parallel).

If they just intersect at a point, then the gradients of the lines would be different, but the y-intercepts could be the same or different (eg. y = 4x + 2 and y = 9x + 2 intersect at one point).

For them to be coincident however, both the gradient and y-intercept must be the same as only then would they be the same line.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

answer would be 8.64 or s = 11π/4 inches

step-by-step explanation:

s = r θ  

where s = arc length  

r = radius = 3 inches  

θ = central angle in radians = ( π /180)*165  

s = 3(π/180)*165  

= 8.64 inches