Question

If \(f(x)=\frac{2 x-B}{3 x+4}\) and \(f(2)=\frac{1}{2},\) what is the value of \(B ?\)

Short answer

The value of B is -1.

Step-by-step solution

- Substitute the Given Value

Given the function is \( f(x) = \frac{2x - B}{3x + 4} \), and we know \( f(2) = \frac{1}{2} \). Let's substitute \( x = 2 \) into the equation so that it matches the given function value. This produces the equation \( f(2) = \frac{2(2) - B}{3(2) + 4} = \frac{4 - B}{6 + 4} \).

- Simplify the Equation

From Step 1, we have the equation \( \frac{4 - B}{10} = \frac{1}{2} \). To eliminate the fraction, multiply both sides by 10: \[ 4 - B = 5 \]

- Solve for B

Now, solve for \( B \) by isolating \( B \). Subtract 4 from both sides:\[ -B = 1 \]Then multiply both sides by -1 to get \( B = -1 \).

Key concepts

Function Evaluation

Function evaluation involves determining the output of a function for a specific input value. In this exercise, we had the function \( f(x) = \frac{2x - B}{3x + 4} \) and we needed to determine \( f(2) \). We were given that \( f(2) = \frac{1}{2} \). To evaluate the function, we simply substituted \( x = 2 \) into the equation:

\( f(2) = \frac{2(2) - B}{3(2) + 4} \).

This substitution leads to the simplified equation of \( \frac{4 - B}{10} \). Function evaluation helps us understand how changes in the input variable affect the output value. This forms the basis for solving equations and understanding the behavior of functions.

Solving Rational Equations

Solving rational equations involves dealing with fractions. In such equations, variables appear in the numerator, the denominator, or both. Our goal is to manipulate the equation to isolate the variable. In this exercise, we simplified the given function \( f(x) \) and arrived at \( \frac{4 - B}{10} = \frac{1}{2} \).

To clear the fraction, we multiplied both sides of the equation by 10:

\[ 10 \times \frac{4 - B}{10} = 10 \times \frac{1}{2} \]

This results in the simpler equation:

\(4 - B = 5 \).

By eliminating the fractions early on, the equation becomes easier to handle. Efficiently simplifying rational equations is key to solving complex algebraic problems.

Isolation of Variable

The isolation of a variable pertains to solving for an unknown by performing arithmetic operations. The goal is to get the variable on one side of the equation by itself. In this exercise, we dealt with the simplified equation:

\( 4 - B = 5 \).

To isolate \( B \), we first subtracted 4 from both sides:

\( 4 - B - 4 = 5 - 4 \), which simplifies to:

\( -B = 1 \).

Next, we multiplied both sides by -1 to solve for \( B \):

\( B = -1 \).

Isolating the variable is an essential algebraic skill, ensuring we can solve for unknowns accurately and confidently. The steps involved are crucial in the broader context of algebraic problem-solving.