Problem 95 - College Algebra

Question

Find \(b\) such that \(\frac{4 x-b}{x-5}=3\) will have a solution set given by \(\varnothing\)

Step-by-Step Solution

Answer

The value for \(b\) so that no solution exists for the equation is \(b \neq 20\).

Step 1: Understand and Analyse the Problem

The condition for this equation to have no solution is when the denominator \(x-5\) equals zero while the numerator \(4x - b\) does not. This results in an undefined expression. So we first solve \(x-5=0\) to find the value of \(x\).

Step 2: Find the Value of \(x\)

Solve \(x - 5 = 0\) for \(x\), by adding 5 to both sides of the equation to get: \(x = 5\).

Step 3: Substitute and Solve for \(b\)

Now we substitute \(x = 5\) into the numerator of the original equation. We get \(4x - b \neq 0\), which simplifies to \(4*5 - b \neq 0\). After simplifying further, we get \(20 - b \neq 0\), and finally on isolating \(b\), we find that \(b \neq 20\).

Key Concepts

Undefined Expressions in Algebra

When solving algebraic equations, we frequently encounter expressions that can become undefined under certain conditions. An undefined expression in algebra occurs when a number is divided by zero. Since division by zero is undefined, any rational expression with a denominator that could potentially be zero must be handled with care.

In the context of rational equations, an important step is to identify the values which make the expression undefined. These are the values that set the denominator to zero. For instance, in the equation \(\frac{4x-b}{x-5}=3\), the denominator is \(x-5\). If we substitute \(x=5\), the denominator becomes zero, and the expression becomes undefined. It is critical to single out such values because they are not included in the solution set, and working with them could lead to incorrect or meaningless results.

Finding No Solution Set

The concept of having 'no solution set', or the empty set \(\varnothing\), in algebra implies there are no real numbers that satisfy an equation. An equation might have no solution when the expressions involved can never be equal, regardless of the values substituted for the variables involved.

For the given rational equation \(\frac{4x-b}{x-5}=3\), we anticipate a scenario where there is no solution. This occurs if the numerator cannot equal zero when the denominator does. In order to have no solution, \(b\) must be chosen such that \(4x - b \eq 0\) whenever \(x-5 = 0\). We identify that when \(x=5\), the denominator is zero, which means that for no solution to exist, the numerator \(4*5 - b\) must not be equal to zero, leading us to the conclusion that \(b\) cannot be equal to 20.

Isolating Variables

Isolating variables is a fundamental technique in algebra used to solve for a specific variable. The goal is to manipulate the equation such that the variable of interest is on one side of the equation and everything else is on the other side.

In the step by step solution, to isolate \(b\) in the equation \(4*5 - b \eq 0\), we keep \(b\) on one side and relocate all other terms to the opposite side. This involves keeping the variable term on the left and moving the constant term to the right by adding or subtracting. Here, we add \(b\) to both sides and then subtract \(20\) from both sides to isolate \(b\), resulting in the inequality \(b \eq 20\). Isolating variables not only helps in solving equations but also allows us to understand the relationships between different variables in an equation.