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Difficulty: MediumSlope of a Line

In the standard (x,y)(x, y) coordinate plane, a line passes through the points (a,2)(a, 2) and (10,a−1)(10, a - 1). If the slope of the line is −13-\frac{1}{3}, what is the value of aa?

Answer: -0.5

Answer

The value of aa is −0.5-0.5 (or −12-\frac{1}{2})
Applying the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} to the points (a,2)(a, 2) and (10,a−1)(10, a - 1) with slope −13-\frac{1}{3} yields the equation −13=a−310−a-\frac{1}{3} = \frac{a - 3}{10 - a}. Solving this linear equation correctly yields a=−0.5a = -0.5.

Step-by-Step Solution

1
Apply the slope formula using the given coordinates.
−13=(a−1)−210−a-\frac{1}{3} = \frac{(a - 1) - 2}{10 - a}
The slope of a line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is defined as m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
2
Simplify the numerator.
−13=a−310−a-\frac{1}{3} = \frac{a - 3}{10 - a}
Subtracting 22 from a−1a - 1 simplifies the numerator to a−3a - 3.
3
Cross-multiply to solve the rational equation.
−1(10−a)=3(a−3)-1(10 - a) = 3(a - 3)
Multiplying both sides by the denominators eliminates the fractions.
4
Solve the linear equation for aa.
a=−0.5a = -0.5
Distributing on both sides gives −10+a=3a−9-10 + a = 3a - 9. Rearranging terms yields 2a=−12a = -1, which simplifies to a=−0.5a = -0.5.

Key Concept

Slope of a Line
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Topics

Slope of a Line

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