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Solving equations is a fundamental aspect of algebra that involves finding the values that satisfy the given mathematical statement. In our specific exercise, the task is to determine a value of \(k\) such that the point \((k, 1)\) satisfies the linear equation \(y = 4x + 8\).
To achieve this, we plug in the coordinates of the point into the equation. This replaces the \(x\) and \(y\) in the equation, allowing us to transform a general statement into one that can be solved for the unknown variable, \(k\).
Solving equations often requires manipulating the equation by performing operations such as addition, subtraction, multiplication, or division to both sides of the equation until the unknown is isolated. In this exercise, substituting the coordinates into the equation is just the start. The next critical step is simplification to continue solving for the unknown value.

The substitution method is a highly effective technique in solving equations, especially when faced with systems of equations or specific values. In this exercise, substitution involves inserting the known values of a point into the given equation.
This method is popularly used for its straightforwardness:

• Replace variables with known values to simplify the equation.
• Reduce the equation to one involving a single unknown, making it easier to solve.

The substitution method allows a complex problem to be addressed more directly by making use of given conditions. In our example, substituting \(k\) for \(x\) and 1 for \(y\) gives us a clear path forward to solve for \(k\), transforming the equation into \(1 = 4k + 8\).
With substitution, the focus shifts from managing several variables to dealing with one, significantly simplifying the solving process.

Algebraic manipulation refers to rearranging and simplifying an equation to find the desired variable. After using substitution to insert \(k\) and \(y=1\) into the equation, we reach the simplified equation \(1 = 4k + 8\).
Algebraic manipulation proceeds from here:

• Subtract 8 from both sides: \(1 - 8 = 4k\) simplifies to \(-7 = 4k\).
• Finally, to isolate \(k\), divide both sides by 4, resulting in \(k = -\frac{7}{4}\).

Through these straightforward steps, we transform the original equation into a solution. Each operation keeps the equation balanced, ensuring that the relationships between numbers remain true.
Remember, algebraic manipulation is about clarity and logic, allowing each transformation to bring you closer to the answer. Whether it’s a simple or a complex equation, mastering these manipulations is crucial for solving algebraic problems efficiently.