Question

the points (k, 4) and (1, - 5) fall on a line with a slope of 9. what is the value of k?

Explanation:

Step1: Recall slope - formula

The slope formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1,y_1)=(k,4)\), \((x_2,y_2)=(1, - 5)\) and \(m = 9\).

Step2: Substitute values into slope - formula

Substitute into \(m=\frac{y_2 - y_1}{x_2 - x_1}\), we get \(9=\frac{-5 - 4}{1 - k}\).

Step3: Simplify the numerator

First, simplify the numerator: \(-5-4=-9\). So the equation becomes \(9=\frac{-9}{1 - k}\).

Step4: Cross - multiply

Cross - multiply to get \(9(1 - k)=-9\).

Step5: Expand the left - hand side

Expand \(9(1 - k)\) to \(9-9k=-9\).

Step6: Solve for \(k\)

Subtract 9 from both sides: \(-9k=-9 - 9=-18\). Then divide both sides by \(-9\), we get \(k = 2\).

Answer:

2