Problem 22 - Algebra Form and Function

Question

If \(h(x)=3-2 / x,\) solve \(3 h(x)+1=7\).

Step-by-Step Solution

Answer

Answer: The solution for the equation is \(x = 2\).

Step 1: Substitute the given function

Replace \(h(x)\) with the given expression \(3 - \frac{2}{x}\) in the equation: $$3(3 - \frac{2}{x}) + 1 = 7$$

Step 2: Distribute and simplify

Distribute the 3 to both terms inside the parentheses and simplify the expression: $$9 - \frac{6}{x} + 1 = 7$$

Step 3: Combine like terms

Combine the constants 9 and 1: $$10 - \frac{6}{x} = 7$$

Step 4: Isolate the fraction term

Subtract 10 from both sides of the equation: $$-\frac{6}{x} = -3$$

Step 5: Solve for x

To solve for x, we need to get rid of the fraction. We can do this by multiplying both sides of the equation by x: $$-6 = -3x$$ Now, divide both sides of the equation by -3: $$x = \frac{-6}{-3}$$

Step 6: Simplify the expression

Simplify the expression to find the value of x: $$x = 2$$ So, the solution to the equation \(3h(x) + 1 = 7\) is \(x = 2\).

Key Concepts

Functions

Functions are a fundamental concept in algebra that describe a relationship between a set of inputs and a set of possible outputs. In any function, each input is paired with exactly one output. For the given problem, the function is defined as \( h(x) = 3 - \frac{2}{x} \). Here, \( h(x) \) is the output which is determined by the input \( x \).
Understanding functions involves knowing how to read and interpret them:

• \( x \) is the variable or the input value.
• \( h(x) \) represents the function value or the output.

Each input \( x \) when plugged into the expression \( 3 - \frac{2}{x} \) will produce a specific value for \( h(x) \).
Functions like \( h(x) = 3 - \frac{2}{x} \) can be recalculated for different values of \( x \). Handling them is similar to how one manipulates standard equations, following rules of arithmetic and algebra.

Substitution

Substitution is the technique used to replace variables, expressions, or functions in equations with their actual values or expressions. It's a critical step in solving equations like the one in the exercise.
To solve \( 3h(x) + 1 = 7 \),

• First, substitute the function definition \( h(x) = 3 - \frac{2}{x} \) directly into the equation.
• This takes the form of replacing every mention of \( h(x) \) in the equation with the expression \( 3 - \frac{2}{x} \).

This transforms the original equation into \( 3(3 - \frac{2}{x}) + 1 = 7 \), which can then be solved using algebraic methods.
Substitution simplifies the handling of complex expressions and allows us to focus on solving the resulting equation with familiar algebraic techniques.

Equation Simplification

Equation simplification is the process of transforming an equation into its simplest form, making it easier to solve. In the exercise, simplifying the equation was necessary after substitution.
Here’s how simplification works in steps:

• Distribute any constants across terms inside parentheses, as in \( 3(3 - \frac{2}{x}) \).
• Combine like terms, such as adding \( 9 \) and \( 1 \) to get \( 10 \).
• Isolate variable terms to one side to simplify further manipulation of the equation.
• Eliminate fractions or complex terms, enabling straightforward solution methods.

In the given problem, simplifying calculations resulted in the equation \( -\frac{6}{x} = -3 \). Then, solving for \( x \) became much more manageable by isolating terms and simplifying further, eventually arriving at \( x = 2 \).
Equation simplification is all about reducing complexity, allowing us to solve for variables efficiently.