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Difficulty: Very hardLinear Equations and Graphing

In the standard (x,y)(x,y) coordinate plane, a line with a negative slope passes through the point (3,2)(3, 2) and has positive integer intercepts (a,0)(a, 0) and (0,b)(0, b). What is the sum of all possible values of aa?

  1. A
    10
  2. B
    15
  3. 24Answer
  4. D
    20
  5. E
    27

Answer

The sum of all possible values of aa is 24.
The correct answer is the sum 24. Writing the line's equation in intercept form gives xa+yb=1\frac{x}{a} + \frac{y}{b} = 1. Substituting the point (3,2)(3, 2) yields 3a+2b=1\frac{3}{a} + \frac{2}{b} = 1. Solving for bb results in b=2+6a−3b = 2 + \frac{6}{a-3}. Because both aa and bb must be positive integers, the expression a−3a-3 must be a positive divisor of 66. The positive divisors of 66 are 1,2,3,1, 2, 3, and 66, which correspond to the aa values of 4,5,6,4, 5, 6, and 99. Summing these values gives 4+5+6+9=244 + 5 + 6 + 9 = 24.

Step-by-Step Solution

1
Set up the intercept form of the linear equation.
xa+yb=1\frac{x}{a} + \frac{y}{b} = 1
A line with xx-intercept (a,0)(a,0) and yy-intercept (0,b)(0,b) can be written in intercept form.
2
Substitute the given point (3,2)(3, 2) into the equation.
3a+2b=1\frac{3}{a} + \frac{2}{b} = 1
Since the line passes through (3,2)(3,2), these coordinates must satisfy the equation.
3
Solve the equation for bb in terms of aa.
2b=1−3a  ⟹  2b=a−3a  ⟹  b=2aa−3\frac{2}{b} = 1 - \frac{3}{a} \implies \frac{2}{b} = \frac{a-3}{a} \implies b = \frac{2a}{a-3}
Expressing bb in terms of aa helps analyze the integer constraints.
4
Rewrite the expression for bb to isolate the fractional part.
b=2(a−3)+6a−3=2+6a−3b = \frac{2(a-3) + 6}{a-3} = 2 + \frac{6}{a-3}
This form allows us to see when bb will be an integer based on the divisors of the numerator.
5
Determine the positive integer solutions for aa and bb.
a−3a-3 must be a positive divisor of 66. The positive divisors of 66 are 1,2,3,1, 2, 3, and 66. This yields:
- If a−3=1  ⟹  a=4,b=8a-3=1 \implies a=4, b=8
- If a−3=2  ⟹  a=5,b=5a-3=2 \implies a=5, b=5
- If a−3=3  ⟹  a=6,b=4a-3=3 \implies a=6, b=4
- If a−3=6  ⟹  a=9,b=3a-3=6 \implies a=9, b=3
Since aa and bb must be positive integers, a−3a-3 must be positive and divide 6 evenly. Negative divisors (like −1-1 or −2-2) would make bb negative or zero.
6
Sum the possible values of aa.
4+5+6+9=244 + 5 + 6 + 9 = 24
To find the final answer, we sum all the valid xx-intercept values.

Key Concept

Using the intercept form of a linear equation and applying integer constraints to find coordinates.
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Topics

Linear Equations and Graphing

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