Problem 100 - Beginning and Intermediate Algebra

Question

If \(f(x)=\frac{1}{3} x+5,\) find \(x\) so that \(f(x)=7\)

Step-by-Step Solution

Answer

The value of \(x\) that makes \(f(x) = 7\) is \(x = 6\).

Step 1: Substitute f(x) with 7

Given that \(f(x) = \frac{1}{3}x + 5\) and \(f(x) = 7\), we can substitute the function with the value 7: \[ 7 = \frac{1}{3}x + 5 \]

Step 2: Isolate x

Now we need to solve the equation for \(x\). Start by subtracting 5 from both sides: \[ 7 - 5 = \frac{1}{3}x + 5 - 5 \] \[ 2 = \frac{1}{3}x \]

Step 3: Solve for x

To solve for \(x\), multiply both sides of the equation by 3: \[ 3(2) = 3\left(\frac{1}{3}x\right) \] \[ 6 = x \]

Step 4: Check the solution

Plug the value of \(x\) back into the original function to ensure it gives the desired output of 7: \[ f(6) = \frac{1}{3}(6) + 5 \] \[ f(6) = 2 + 5 \] \[ f(6) = 7 \] Since the output is 7, the solution is correct. The value of \(x\) that makes \(f(x) = 7\) is \(x = 6\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in algebra that concerns finding the output of a function for a specified input value. When we have a function like \( f(x) = \frac{1}{3}x + 5 \), the process of function evaluation involves determining \( f(x) \) when a particular value is plugged in for \( x \).

• Step 1: Identify the function and the value for \( f(x) \).


• Step 2: Substitute the specific value in place of \( x \).


• Step 3: Calculate the result.

Using our given exercise, we substituted \( f(x) = 7 \), so the function becomes \( 7 = \frac{1}{3}x + 5 \). This sets the stage for solving an equation where we know the result \( f(x) \) and need to find what input value \( x \) produces that result. By evaluating functions, we can solve problems where the connection between input and output needs clarification.

Isolating Variables

Isolating variables is a critical skill in algebra used to solve equations. It involves manipulating an equation so that a particular variable stands alone on one side of the equation. This helps determine the value of that variable. When isolating variables, it's crucial to perform the same operation on both sides of the equation to maintain its balance and equality.

• Begin by identifying terms that need to be moved to the opposite side.


• Add or subtract terms to both sides as necessary.


• Multiply or divide to eliminate coefficients.

In the exercise, we isolated \( x \) by first subtracting 5 from both sides of the equation \( 7 = \frac{1}{3}x + 5 \), resulting in \( 2 = \frac{1}{3}x \). This isolated the term with \( x \) before solving for it entirely. Isolating variables is a step-by-step process that eventually reveals the variable's value, which is necessary for solving equations.

Solving Equations

Solving equations involves finding the values of variables that make an equation true. This process is essential in algebra and helps us understand relationships between quantities. In solving linear equations, we use a combination of arithmetic operations to simplify and find the required variable.

• Simplify expressions on both sides of the equation as much as possible.


• Use inverse operations to undo any addition, subtraction, multiplication, or division affecting the variable.


• Check the solution by substituting it back into the original equation.

In our example, after isolating \( x \), we multiply both sides by 3 to solve \( 2 = \frac{1}{3}x \), resulting finally in \( 6 = x \). Solving the equation means determining and verifying that \( x = 6 \) is indeed the input for which \( f(x) = 7 \). Thus, solving equations is about finding values and understanding their roles in mathematical relationships.