Problem 94 - College Algebra

Question

Find \(b\) such that \(\frac{7 x+4}{b}+13=x\) will have a solution set given by \(\\{-6\\}\)

Step-by-Step Solution

Answer

The value of \(b\) that makes the equation \(\frac{7x + 4}{b} + 13 = x\) have a solution set of \{-6\}, is \(b = 2\).

Step 1: Substitute the value of x into the equation

We know from the given solution set that \(x = -6\). Substitute \(x\) with -6 into the original equation, we get: \(\frac{7*(-6) + 4}{b} + 13 = -6\).

Step 2: Simplify the equation

When we simplify the equation, we end up with \(\frac{-38}{b} + 13 = -6\).

Step 3: Isolate b

This requires first removing the 13 from the left side by subtraction, so that it moves to the right side of the equation and becomes 19 (i.e., -6 - 13 = -19). The equation will then become \(\frac{-38}{b} = -19\). To isolate \(b\), you can simply cross-multiply to get \(b = \frac{-38}{-19}\).

Step 4: Calculate the value of b

Divide -38 by -19, we find that \(b = 2\).