Question

Find the value of \(k\) so that the line through the given points has slope \(m .\) $$ (k, k+1),(3,2) ; m=3 $$

Short answer

The value of \(k\) that makes the slope of the line through the points \((k, k+1)\) and \((3,2)\) equal to 3 is \(k = 4\).

Step-by-step solution

Write down and simplify the slope equation

We know that the slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). We will substitute \(m = 3\), \(x_1 = k\), \(y_1 = k + 1\), \(x_2 = 3\), and \(y_2 = 2\) into this formula. Doing this we get a simplified equation, which is \(3 = \frac{2-(k+1)}{3-k}\)

Solve the equation for 'k'

First, cross multiply to remove the denominator and get rid of the fraction. This gives: \(3(3-k) = 2-(k+1)\). Then, distribute the 3 into the (3-k) to give: \(9 - 3k = 2 - k - 1\). After simplifying that equation, we get: \(3k - k = 9 - 2 + 1\). Further simplifying gives: \(2k = 8\). Solving for 'k', we get \(k = \frac{8}{2} = 4\)

Check the solution

We now need to verify if \(k = 4\) actually satisfies the original equation. Substituting \(k = 4\) in the original equation shows that both sides are equal, hence, confirming that our solution is correct.