91Ó°ÊÓ

Function evaluation is the process of determining the output of a function for a given input. In the problem, the function is defined as \( f(x) = -2x + c \). To evaluate this function, we substitute a specific value for \( x \).

In our exercise, we need to evaluate \( f(3) \). This means we use 3 as the input value into the function. When we substitute 3 for \( x \), the expression becomes \( -2(3) + c \).

Evaluating a function is like giving it a specific test number to see what happens. Think of it as plugging a number into a machine and observing the result. In this case, after evaluating, we have the expression \( -6 + c \), which sets the stage for solving the rest of the equation.

Solving equations involves finding the value of unknown variables that make the equation true. In our example, the equation that needs to be solved is formed after substituting \( x = 3 \) into the function evaluation.

Given that \( f(3) = 1 \), we set \(-6 + c = 1\). This equation represents a balance, where both sides must be equal. Here, solving the equation translates to finding the appropriate value of \( c \) such that the equation holds true.

To solve for \( c \), we rearrange the equation by isolating \( c \) on one side. This typically involves simple algebraic manipulations such as adding or subtracting numbers. Here, we add 6 to both sides, a method from basic algebra called balancing or maintaining equality, which leads to the solution \( c = 7 \).

This process is not only critical in algebra courses but also across many fields where equations are used to model relationships and solve problems.

The substitution method is a key algebraic technique used to simplify and solve equations or systems of equations. It involves replacing one variable with its corresponding known value or expression.

In our exercise, the substitution method was used to replace \( x \) with 3 in the function \( f(x) \). This simple substitution transforms the function equation into \( f(3) = -2(3) + c \), allowing us to work with specific numbers rather than variables.

By substituting known values into an equation, we can often simplify the problem and solve for other unknowns. This method can also be used in more complex systems of equations, not just single-variable ones, providing a powerful tool for solving many mathematical problems.

• Identify a substitution point – a known value for a variable.
• Replace the variable in the equation with this value.
• Simplify the equation and solve for the unknown.

The substitution method is fundamental in many areas of math and science, allowing simplifications that make complex problems manageable.