In the standard (x, y) coordinate plane, if the slope of a line passing through the points… — ACT

In the standard $(x, y)$ coordinate plane, if the slope of a line passing through the points $(k, k^2)$ and $(k+1, (k+1)^2)$ is an integer, then $k$ must be an integer. Is this statement true or false?

Answer: Answer

Answer

False

The statement is false because the simplified slope of the line is

2k + 1

. The slope

2k + 1

is an integer for any half-integer value of

k

(such as

k = 0.5

, which yields a slope of

2

), showing that

k

does not have to be an integer.

Step-by-Step Solution

Use the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

with the given coordinates

(k, k^2)

and

(k+1, (k+1)^2)

The slope is

m = \frac{(k+1)^2 - k^2}{(k+1) - k}

To express the slope of the line in terms of the variable

k

Simplify the numerator and denominator of the slope expression.

m = \frac{k^2 + 2k + 1 - k^2}{1} = 2k + 1

To obtain a simplified algebraic expression for the slope.

Set the slope equal to an arbitrary integer

n

and solve for

k

2k + 1 = n \implies k = \frac{n-1}{2}

To find the set of values for

k

that yield an integer slope.

Evaluate if

k

must be an integer by choosing an integer value for

n

n = 2

, then

k = \frac{2-1}{2} = 0.5

, which is not an integer.

To test the truth value of the statement by finding a counterexample.

Key Concept

Determining the slope of a line from two coordinate points and analyzing the algebraic properties of the slope expression.

Question

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Step-by-Step Solution

Key Concept

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