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Difficulty: MediumSlope of a Line

A line graphed on the standard (x,y)(x, y) coordinate plane has a slope of 0.50.5 and passes through the points (p,3)(p, 3) and (7,2p−1)(7, 2p - 1). What is the value of pp?

Answer: 3

Answer

The value of pp is 33.
Applying the slope formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} to the points (p,3)(p, 3) and (7,2p−1)(7, 2p - 1) with m=0.5m = 0.5 yields the linear equation 0.5=2p−47−p0.5 = \frac{2p - 4}{7 - p}. Cross-multiplying and solving for pp gives p=3p = 3.

Step-by-Step Solution

1
Set up the slope formula using the given points and slope.
0.5=(2p−1)−37−p0.5 = \frac{(2p - 1) - 3}{7 - p}
The slope of a line passing through (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
2
Simplify the numerator of the fraction.
0.5=2p−47−p0.5 = \frac{2p - 4}{7 - p}
Combine the constant terms −1-1 and −3-3 in the numerator.
3
Multiply both sides by the denominator to clear the fraction.
0.5(7−p)=2p−40.5(7 - p) = 2p - 4
Eliminating the denominator allows solving the linear equation.
4
Distribute the 0.50.5 on the left side of the equation.
3.5−0.5p=2p−43.5 - 0.5p = 2p - 4
Multiply each term inside the parentheses by 0.50.5.
5
Isolate the variable pp by rearranging terms.
2.5p=7.52.5p = 7.5
Add 0.5p0.5p to both sides and add 44 to both sides.
6
Divide by the coefficient of pp to solve for pp.
p=3p = 3
Dividing 7.57.5 by 2.52.5 yields the final value of the parameter.

Key Concept

Calculating a parameter using the slope formula

Alternative Method

Instead of using the decimal 0.50.5, write it as the fraction 12\frac{1}{2}. The equation becomes 12=2p−47−p\frac{1}{2} = \frac{2p - 4}{7 - p}. Cross-multiplying gives 1(7−p)=2(2p−4)⇒7−p=4p−8⇒15=5p⇒p=31(7 - p) = 2(2p - 4) \Rightarrow 7 - p = 4p - 8 \Rightarrow 15 = 5p \Rightarrow p = 3.
Estimated Time:1m 30s
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Topics

Slope of a Line

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