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Difficulty: MediumLinear Equations and Graphing

In the standard (x,y)(x, y) coordinate plane, a line LL passes through the point (2,3)(2, 3) and has a positive yy-intercept (0,b)(0, b). The area of the triangular region in the first quadrant bounded by line LL, the xx-axis, and the yy-axis is 1212 square units. What is the value of bb?

  1. A
    33
  2. B
    44
  3. 66Answer
  4. D
    88
  5. E
    1212

Answer

The correct value of bb is 66.
To find the value of bb, we can use the intercept form of a linear equation: xa+yb=1\frac{x}{a} + \frac{y}{b} = 1, where aa is the xx-intercept and bb is the yy-intercept. The area of the right triangle formed by the axes and the line is given by 12ab=12\frac{1}{2}ab = 12, which means ab=24ab = 24, or a=24ba = \frac{24}{b}. Since the line passes through the point (2,3)(2, 3), we substitute these coordinates into the intercept equation to get 2a+3b=1\frac{2}{a} + \frac{3}{b} = 1. Substituting a=24ba = \frac{24}{b} into this equation gives 2b24+3b=1\frac{2b}{24} + \frac{3}{b} = 1, which simplifies to b12+3b=1\frac{b}{12} + \frac{3}{b} = 1. Multiplying the entire equation by 12b12b to clear the denominators results in b2+36=12bb^2 + 36 = 12b. Rearranging this quadratic equation gives b2−12b+36=0b^2 - 12b + 36 = 0, which factors as (b−6)2=0(b - 6)^2 = 0. Solving for bb yields b=6b = 6.

Step-by-Step Solution

1
Express the area of the right triangle in the first quadrant in terms of the xx-intercept (a,0)(a, 0) and yy-intercept (0,b)(0, b).
The area is 12ab=12\frac{1}{2}ab = 12, which simplifies to ab=24ab = 24, or a=24ba = \frac{24}{b}.
The triangular region is a right triangle with base aa and height bb along the coordinate axes.
2
Set up the equation of the line using the intercept form xa+yb=1\frac{x}{a} + \frac{y}{b} = 1 and substitute the given point (2,3)(2, 3).
Substituting x=2x = 2 and y=3y = 3 gives the equation 2a+3b=1\frac{2}{a} + \frac{3}{b} = 1.
Since the line passes through the point (2,3)(2, 3), this point must satisfy the equation of the line.
3
Substitute a=24ba = \frac{24}{b} into the intercept equation and solve the resulting quadratic equation for bb.
Substituting aa gives 224b+3b=1  ⟹  b12+3b=1\frac{2}{\frac{24}{b}} + \frac{3}{b} = 1 \implies \frac{b}{12} + \frac{3}{b} = 1. Multiplying by 12b12b yields b2+36=12b  ⟹  b2−12b+36=0  ⟹  (b−6)2=0  ⟹  b=6b^2 + 36 = 12b \implies b^2 - 12b + 36 = 0 \implies (b - 6)^2 = 0 \implies b = 6.
Solving the quadratic equation yields the value of the yy-intercept bb.

Key Concept

Linear equations and graphing using intercept form and triangle area relations

Alternative Method

An alternative approach is to use the slope formula. The line passes through (a,0)(a, 0), (2,3)(2, 3), and (0,b)(0, b). The slope between (0,b)(0, b) and (2,3)(2, 3) is 3−b2\frac{3 - b}{2}, and the slope between (0,b)(0, b) and (a,0)(a, 0) is −ba-\frac{b}{a}. Equating these gives 3−b2=−ba  ⟹  3a−ab=−2b\frac{3 - b}{2} = -\frac{b}{a} \implies 3a - ab = -2b. Since the area is 1212, we know ab=24ab = 24. Substituting ab=24ab = 24 gives 3a−24=−2b  ⟹  3a+2b=243a - 24 = -2b \implies 3a + 2b = 24. Since a=24ba = \frac{24}{b}, we get 3(24b)+2b=24  ⟹  72b+2b=24  ⟹  b2−12b+36=0  ⟹  b=63\left(\frac{24}{b}\right) + 2b = 24 \implies \frac{72}{b} + 2b = 24 \implies b^2 - 12b + 36 = 0 \implies b = 6.
Estimated Time:1m 30s
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Topics

Linear Equations and Graphing

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