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Linear equations like the one we're working with here, \( h(x) = 3x - 5 \), are the building blocks of algebra. They're called linear because they represent straight lines when you graph them on a coordinate plane. In general, a linear equation can be written as \( y = mx + b \). Here, \( m \) is the slope, and \( b \) is the y-intercept. The slope indicates how steep the line is, and the y-intercept tells us where the line crosses the y-axis. To grasp linear equations fully, always pay attention to:

• Slope (m): Tells you how the line "rises" or "runs". A positive slope means the line rises from left to right, while a negative slope falls.
• Y-intercept (b): This is the starting point of your line when \( x = 0 \).

Breaking down equations with these parameters makes it easier to plot or interpret any line on a graph. For example, in \( h(x) = 3x - 5 \), the slope \( m \) is 3, indicating a moderate upward rise. The y-intercept, \( -5 \), signifies where our line crosses the y-axis.

The y-intercept is a key concept of linear equations that can sometimes be confusing. Simply put, it's where the line crosses the y-axis. To find it, all you have to do is set \( x = 0 \) in the equation. This is because the y-axis is the vertical line where \( x \) values are always zero.For instance, in the equation \( h(x) = 3x - 5 \), finding the y-intercept involves substituting 0 for \( x \):\[ h(0) = 3(0) - 5 = -5 \]This calculation tells us that the y-intercept is \( (0, -5) \).
The operations are straightforward:

• Substitution: Replace \( x \) with 0.
• Simplification: Solve to find \( y \).

Understanding the y-intercept helps you quickly identify where the line starts on the y-axis, providing a foundation for sketching or analyzing the line's behavior further.

The x-intercept is the point where the graph crosses the x-axis. At this point, the value of \( y \) or \( h(x) \) is always zero because the point is on the horizontal line where all \( y \) values are zero. To find the x-intercept of a linear equation, you set the output to zero and solve for \( x \).Let's explore this with our equation \( h(x) = 3x - 5 \):1. Set \( h(x) \) to 0: \[ 0 = 3x - 5 \]2. Solve for \( x \):

• Add 5 to both sides: \( 5 = 3x \)
• Divide both sides by 3: \( x = \frac{5}{3} \)

Therefore, the x-intercept is at \( \left( \frac{5}{3}, 0 \right) \).
This procedure reveals where the line hits the x-axis, a horizontal reference in your graph. The x-intercept provides insight into how a line situates itself on the coordinate plane, revealing its course and intersection points.