In the standard (x, y) coordinate plane, a line with a slope of \frac{3}{2} passes through… — ACT

In the standard $(x, y)$ coordinate plane, a line with a slope of $\frac{3}{2}$ passes through the points $(k, -2)$ and $(2k, 4)$ . Is the statement ' $k = 4$ ' true or false?

Answer: Answer

Answer

The statement is true because solving the slope equation with the given coordinates and slope value yields

k = 4

The correct option is true because substituting

k = 4

into the coordinates yields the points

(4, -2)

and

(8, 4)

. Using the slope formula, the slope is calculated as

\frac{4 - (-2)}{8 - 4} = \frac{6}{4} = \frac{3}{2}

, which matches the given slope value.

Step-by-Step Solution

State the formula for the slope

m

of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

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

This formula defines the constant rate of change (slope) between any two points on a non-vertical line.

Substitute the coordinates

(k, -2)

and

(2k, 4)

and the slope value

\frac{3}{2}

into the formula.

\\frac{3}{2} = \frac{4 - (-2)}{2k - k}

This creates an algebraic equation in terms of

k

that represents the given geometric conditions.

Simplify the numerator and denominator of the fraction and solve the resulting equation for

k

\frac{3}{2} = \frac{6}{k} \Rightarrow 3k = 12 \Rightarrow k = 4

Simplification reduces the algebraic expression, allowing the variable

k

to be isolated and solved.

Key Concept

Calculating the slope of a line from two coordinate points containing an algebraic parameter.

Estimated Time:1m 30s

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