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The vertical intercept of a linear function is the point where the graph of the function crosses the vertical (y-axis). This occurs when the input value, often represented by \(x\), is set to zero. It's essentially the value of the function when no horizontal movement is considered.
Here's how you can easily find it for the function \(k(x) = -5x + 1\):

• Set \(x = 0\).
• Solve for \(k(x)\), which results in \(k(0) = -5(0) + 1\).
• Simplify to find \(k(0) = 1\).

Therefore, the vertical intercept of this function is at the point \((0, 1)\). In more general terms, for any equation in the form \(y = mx + b\), the vertical intercept will simply be the constant value \(b\). This is because evaluating the function at zero essentially leaves us with this constant term.

The horizontal intercept of a linear function is where the function's graph touches the horizontal (x-axis). At this point, the output, or \(y\), of the function is zero. Finding the horizontal intercept essentially means solving for \(x\) when \(k(x) = 0\).
Let's break down the steps using the equation \(k(x) = -5x + 1\):

• Set the equation equal to zero: \(-5x + 1 = 0\).
• Subtract 1 from both sides to get \(-5x = -1\).
• Finally, divide by \(-5\), leading to \(x = \frac{1}{5}\).

This tells us the horizontal intercept is at the point \(\left(\frac{1}{5}, 0\right)\). It's helpful to remember that for linear equations of the form \(y = mx + b\), the horizontal intercept is found by solving the equation \(mx + b = 0\). This gives us the point on the x-axis where the line crosses.

Solving linear equations is a key concept in algebra that involves finding the value of the variable that makes the equation true. Linear equations, which form straight lines when graphed, are expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-axis intercept.
To find solutions to linear equations:

• Isolate the variable of interest (usually \(x\) or \(y\)) on one side of the equation.
• This often involves adding, subtracting, multiplying, or dividing both sides of the equation to simplify.

In the case of finding intercepts, you'll either set \(x = 0\) to find the vertical intercept or set the equation to zero to solve for \(x\) for finding the horizontal intercept.
For example, in \(k(x) = -5x + 1\):
- Finding the vertical intercept involves setting \(x = 0\), simplifying gives \(y = 1\).
- The horizontal intercept involves setting \(y = 0\), solve for \(x\) gives \(x = \frac{1}{5}\).
Linear equations are foundational to understanding more complex mathematical models, making these skills crucial for further study in mathematics.