Question 999176: write an equation in standard form of a line whose x-intercept is an integer and y-intercept is a fraction. Explain how you know that the x-intercept is an integer and y-intercept is a fraction
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! ax + by = c
the x-intercept is the value of x when y = 0.
the y-intercept is the value of y when x = 0.
when y = 0, ax = c which results in x = c/a
when x = 0, by = c which results in y = c/b
in order for c/a to be an integer, c has to be a multiple of a.
in order for c/b to be a fraction, c must not be a multiple of b.
if c is a multiple of a, then c can be made equal to a*d, where d is an integer.
example:
55 / 5 is an integer because 55 is a multiple of 5.
55 = 5 * 11, so 55/5 = 11
56 / 5 is not an integer because 56 is not a multiple of 5.
-15 / 5 is an integer because 15 is a multiple of 5.
15 = 5 * 3, so 15/5 = 3
-16 / 5 is not an integer because 16 is not a multiple of 5.
so, if you want ax + by = c to have the x-intercept an integer and the y-intercept a fraction, then c must be a multiple of a but not a multiple of b.
an example would be:
5x + 6y = 15
the x intercept is the value of x when y = 0.
when y = 0, the equation becomes 5x = 15.
solve for x to get x = 15/5 = 3.
3 is an integer.
the y intercept is the value of y when x = 0.
when x = 0, the equation becomes 6y = 15.
solve for y to get y = 15/6.
15/6 is not an integer.
that's my take.
an integer can also be described as a fraction where the denominator is equal to 1.
why is 55/5 an integer?
multiply numerator and denominator of 55/5 by 1/5 and you get (55*1/3) / (5*1/5) which results in 11/1 which is the same as 11.
55/5 is therefore an integer because it can be simplified to a ratio where the denominator is equal to 1.
56/5 cannot be simplified to a ratio where the numerator is an integer and the denominator is equal to 1.
this is because 56 is not a multiple of 5.
you can find a number such that the denominator is 1, but the number will not be an integer.
56/5 = 11.2/1 = 11.2 which is a fraction in decimal form.
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