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Difficulty: MediumLinear Functions and Graphs

A line in the xyxy-plane representing the linear function ff has a yy-intercept of (0,8)(0, 8) and contains the point (a,b)(a, b), where aa and bb are positive integers. If the slope of this line is −35-\frac{3}{5} and the sum of aa and bb is 1212, what is the value of aa?

  1. A
    2
  2. B
    6
  3. C
    8
  4. 10Answer

Answer

10
Since the line representing the function has a y-intercept of (0, 8) and a slope of -3/5, its equation in slope-intercept form is y = -3/5x + 8. Since the line contains the point (a, b), substituting x = a and y = b into this equation gives b = -3/5a + 8. The sum of a and b is given as 12, so a + b = 12, which can be rewritten as b = 12 - a. Substituting this expression for b gives the equation 12 - a = -3/5a + 8. Subtracting 8 from both sides yields 4 - a = -3/5a. Adding a to both sides yields 4 = 2/5a. Multiplying both sides by 5/2 gives a = 10.

Step-by-Step Solution

1
Write the equation of the linear function in slope-intercept form.
y=−35x+8y = -\frac{3}{5}x + 8
The y-intercept of the line is given as (0, 8), and the slope of the line is given as -3/5.
2
Substitute the point (a, b) into the equation of the line.
b=−35a+8b = -\frac{3}{5}a + 8
Since the point (a, b) lies on the line, its coordinates must satisfy the equation of the line.
3
Express b in terms of a using the given sum relationship.
b=12−ab = 12 - a
We are given that the sum of a and b is 12, which can be written as a + b = 12.
4
Substitute the expression for b into the line's equation and solve for a.
a=10a = 10
Substituting b = 12 - a into b = -3/5a + 8 yields 12 - a = -3/5a + 8, which simplifies to 4 = 2/5a, leading to a = 10.

Key Concept

Linear Functions and Graphs
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Topics

Linear Functions and Graphs

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