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Bunuel
How many points on line with equation \(\frac{x}{13}+\frac{y}{52}=1\) have both non-negative integer coordinates?

(A) 10
(B) 12
(C) 13
(D) 14
(E) Infinite

The equation is in form 4x+ y = 52

Since both are non negative integer, x can take values from 0 to 13 (14 values) for which y will always be integer and non negative. y can take 14 values from 52 to 0 in multiple of 4.

Answer is 14 D

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Turn the Intercept Form of the equation of a line into a standard linear indeterminate equation.

52X + 13Y = (13) (52)

Divide both sides of the equation by 13


4X + Y = 52

1st coordinate pairs that we can start with are the Intercepts

(0 , 52)

(13 , 0)

If we have a non negative integer constraint and are given

4X + Y = 52

Starting with the first ordered pair that satisfies the conditions

X = 13 and Y = 0

To find each consecutive integer values that will satisfy the equation,

we can subtract from the X value of 13 the coefficient in front of Y (1)

and

ADD to the Y value of 0 the coefficient in front of X (4)

X = 13 and Y = 0

X = 13 - 1 = 12 —— and —— Y = 0 + 4 = 4

(12 , 4) will be the next pair that satisfies

We can keep following this pattern until we get to the value when X = 0)

(11 , 8)

(10 , 12)

Etc.

(0 , 52)


The number of ordered pairs is the Count of integer terms in the set [0 thru 13] inclusive


14 (d)

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y=-4x+52
y>=0
-4x+52>=0
x<=13
x can be non negative only.
So 0<=x<=13
X can be 0,1,2..13 => 14 integers.
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