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.

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