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Zorluk: OrtaLinear 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 yy-intercept of (0,b)(0, b), where b>0b > 0. If the area of the triangular region bounded by the line LL, the xx-axis, and the yy-axis is 44 square units, what is the value of bb?

  1. A
    2
  2. B
    4
  3. 6Cevap
  4. D
    8
  5. E
    12

Cevap

6
The correct answer is the option containing 6. Using the two points (2,−3)(2, -3) and (0,b)(0, b), we find the slope of the line is m=−b+32m = -\frac{b+3}{2}, which gives the equation y=−b+32x+by = -\frac{b+3}{2}x + b. Setting y=0y=0 shows that the xx-intercept is at x=2bb+3x = \frac{2b}{b+3}. The area of the right triangle formed by the intercepts and the origin is 12×base×height=12×2bb+3×b=b2b+3\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \frac{2b}{b+3} \times b = \frac{b^2}{b+3}. Setting this area equal to 44 yields b2−4b−12=0b^2 - 4b - 12 = 0. Factoring gives (b−6)(b+2)=0(b - 6)(b + 2) = 0. Since we are given b>0b > 0, bb must be 66.

Adım Adım Çözüm

1
Find the slope of the line LL in terms of bb.
The slope mm is given by −3−b2−0=−b+32\frac{-3 - b}{2 - 0} = -\frac{b+3}{2}.
The line passes through (2,−3)(2, -3) and its yy-intercept (0,b)(0, b).
2
Write the equation of the line LL and determine its xx-intercept.
The equation is y=−b+32x+by = -\frac{b+3}{2}x + b. Setting y=0y = 0 gives the xx-intercept x=2bb+3x = \frac{2b}{b+3}.
The xx-intercept is the point where the line crosses the xx-axis (y=0y = 0).
3
Set up the area equation for the triangle formed by the axes and the line.
The area is 12×base×height=12×2bb+3×b=b2b+3=4\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \frac{2b}{b+3} \times b = \frac{b^2}{b+3} = 4.
The base of the triangle is the xx-intercept, and the height is the yy-intercept bb (since both are positive for b>0b > 0).
4
Solve the quadratic equation for bb.
b2=4(b+3)⇒b2−4b−12=0⇒(b−6)(b+2)=0b^2 = 4(b + 3) \Rightarrow b^2 - 4b - 12 = 0 \Rightarrow (b - 6)(b + 2) = 0. Since b>0b > 0, b=6b = 6.
We must solve the equation and select the positive solution because the problem specifies b>0b > 0.

Anahtar Kavram

Using the coordinates of a point and intercepts to write a linear equation, finding intercepts, and calculating the area of a coordinate triangle.

Alternatif Yöntem

Instead of setting up the area algebraically first, you can test the answer choices. For example, testing the correct value 6: the y-intercept is (0,6)(0, 6). The slope of the line passing through (0,6)(0, 6) and (2,−3)(2, -3) is m=−3−62−0=−4.5m = \frac{-3 - 6}{2 - 0} = -4.5. The equation of the line is y=−4.5x+6y = -4.5x + 6. The x-intercept is found by setting y=0y = 0, giving x=64.5=43x = \frac{6}{4.5} = \frac{4}{3}. The area of the triangle is 12×43×6=4\frac{1}{2} \times \frac{4}{3} \times 6 = 4, which matches the given area of 4 square units.
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Linear Equations and Graphing

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