Maths-
SAT
Easy

Question

In the xy-plane, the graph of a linear equation of the form y = mx + b and the graph of an exponential equation of the form y = abx both contain points (1, 3) and (2, 4). If the point (r, s) is on the graph of the linear equation and the point (r, t) is on the graph of the exponential equation, where 0 < r < 4 and s > t, which of the following must be true?

  1. 0 < r < 1
  2. 1 < r < 2
  3. 2 < r < 3
  4. 3 < r < 4

The correct answer is: 3 < r < 4


    y = mx + b is line through (1, 3) & (2, 4)
    fraction numerator y minus y subscript 1 over denominator y subscript 2 minus y subscript 1 end fraction equals fraction numerator x minus x subscript 1 over denominator x subscript 2 minus x subscript 1 end fraction
    fraction numerator y minus 3 over denominator 4 minus 3 end fraction equals fraction numerator x minus 1 over denominator 2 minus 1 end fraction rightwards double arrow y minus 3 equals x minus 1 rightwards double arrow y equals x plus 2
    y equals a b to the power of x text  contains  end text left parenthesis 1 comma 3 right parenthesis & left parenthesis 2 comma 4 right parenthesis
    3 = ab
    4 = ab2 rightwards double arrow 4 over 3 equals b rightwards double arrow 3 equals a 4 over 3
    rightwards double arrow a equals 9 over 4
    (r, s) is on line rightwards double arrows = r + 2.
    (r, t) is on exponential rightwards double arrow t = abr rightwards double arrow t equals 9 over 4 open parentheses 4 over 3 close parentheses to the power of r
    r plus 2 greater than 9 over 4 open parentheses 4 over 3 close parentheses to the power of r.

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