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Zorluk: ZorSlope of a Line

In the standard (x,y)(x, y) coordinate plane, the points (k,−3)(k, -3), (1,k)(1, k), and (4,9)(4, 9) lie on the same straight line, where kk is a constant. Which of the following is a possible value for the slope of this line?

  1. A
    −34-\frac{3}{4}
  2. B
    43\frac{4}{3}
  3. −43-\frac{4}{3}Cevap
  4. D
    44
  5. E
    223\frac{22}{3}

Cevap

The correct answer is −43-\frac{4}{3}.
The correct answer is −43-\frac{4}{3}. If three points are collinear, the slope computed between any two pairs must be equal. Setting the slope between (k,−3)(k, -3) and (1,k)(1, k) equal to the slope between (1,k)(1, k) and (4,9)(4, 9) gives the equation k+31−k=9−k3\frac{k + 3}{1 - k} = \frac{9 - k}{3}. Cross-multiplying yields 3(k+3)=(9−k)(1−k)3(k + 3) = (9 - k)(1 - k), which simplifies to k2−13k=0k^2 - 13k = 0. Solving this gives k=0k = 0 or k=13k = 13. Substituting k=13k = 13 back into the coordinates gives the points (13,−3)(13, -3), (1,13)(1, 13), and (4,9)(4, 9). The slope of the line passing through these points is 9−134−1=−43\frac{9 - 13}{4 - 1} = -\frac{4}{3}.

Adım Adım Çözüm

1
Set up the collinearity condition using the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
The slope of the line segment between (k,−3)(k, -3) and (1,k)(1, k) must equal the slope of the line segment between (1,k)(1, k) and (4,9)(4, 9).
Since all three points lie on the same straight line, the slope between any two pairs of points must be equal.
2
Write the algebraic expressions for the slopes and set them equal to each other.
k−(−3)1−k=9−k4−1⇒k+31−k=9−k3\frac{k - (-3)}{1 - k} = \frac{9 - k}{4 - 1} \Rightarrow \frac{k + 3}{1 - k} = \frac{9 - k}{3}
This establishes a rational equation containing the variable kk.
3
Cross-multiply to eliminate the fractions and simplify.
3(k+3)=(9−k)(1−k)⇒3k+9=9−10k+k23(k + 3) = (9 - k)(1 - k) \Rightarrow 3k + 9 = 9 - 10k + k^2
Cross-multiplying allows us to convert the rational equation into a polynomial equation.
4
Rearrange the terms into standard quadratic form and solve for kk.
k2−13k=0⇒k(k−13)=0⇒k=0 or k=13k^2 - 13k = 0 \Rightarrow k(k - 13) = 0 \Rightarrow k = 0 \text{ or } k = 13
Setting the quadratic expression to zero allows us to find the two possible values for the constant kk by factoring.
5
Calculate the slope of the line for each possible value of kk.
If k=0k = 0, the slope is m=9−04−1=3m = \frac{9 - 0}{4 - 1} = 3. If k=13k = 13, the slope is m=9−134−1=−43m = \frac{9 - 13}{4 - 1} = -\frac{4}{3}.
We must check which of the calculated slopes matches one of the given choices.

Anahtar Kavram

Finding the slope of a line and using the condition of collinearity to solve for missing coordinates.
Tahmini Süre:2m 0s
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Konular

Slope of a Line

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