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The slope is a crucial concept in understanding linear equations. It represents the steepness or incline of a line on a graph. In mathematical terms, the slope is the change in the y-value divided by the change in the x-value, commonly referred to as "rise over run."

For a given equation in the form \( y = mx + c \), the slope is represented by \( m \).
This tells us how much \( y \) will change when \( x \) is increased by one unit.

• If \( m > 0 \), the slope is positive, meaning the line ascends as it moves from left to right.
• If \( m < 0 \), the slope is negative, indicating the line descends as it moves from left to right.
• If \( m = 0 \), the line is perfectly horizontal, showing no change in \( y \) as \( x \) changes.

In our exercise, the first equation \( y = 100 + 5x \) has a slope of 5, meaning it rises by 5 units vertically for each unit it moves horizontally. The second equation \( y = 400 - 4x \) has a slope of -4, descending 4 units in \( y \) for each unit increased in \( x \). Understanding these slopes helps us predict and describe the behavior of the lines.

The y-intercept is the point where a line crosses the y-axis. It is defined by the value of \( y \) when \( x = 0 \). In the linear equation format \( y = mx + c \), the y-intercept is represented by \( c \).

This value provides a starting point on the graph and is crucial for plotting a line.

Consider the equations from the exercise:

• For \( y = 100 + 5x \), the y-intercept is 100. This means the line will intersect the y-axis at the point (0, 100).
• For \( y = 400 - 4x \), the y-intercept is 400. Here, the line intersects the y-axis at the point (0, 400).

Knowing the y-intercept allows us to quickly find where the line begins on the y-axis, making it simpler to draw and interpret linear graphs.

In linear equations, a positive relationship indicates that as the value of one variable increases, the other variable also increases. Specifically, this relationship is present when the slope \( m \) is positive.

In the graph of the equation, you'll see a line that rises upwards from left to right.

This can be easily remembered with these characteristics:

• A line moving upwards reflects that both variables are increasing together.
• Positive slope means each step to the right on the x-axis results in a step up on the y-axis.
• The larger the slope, the steeper the ascent of the line.

In the original exercise, the equation \( y = 100 + 5x \) depicts a positive relationship since the slope is 5. For each unit \( x \) moves to the right, \( y \) increases by 5 units, visually creating an upward-sloping line on the graph.

Conversely, a negative relationship between two variables means that as one variable increases, the other decreases. This scenario occurs when the slope \( m \) is negative in the equation.

On the graph, the line appears to decline as it moves from left to right.

Here's how you can spot and understand negative relationships:

• The line slopes downwards, illustrating that an increase in one variable leads to a decrease in the other.
• Every step to the right on the x-axis results in a step down on the y-axis.
• A larger negative slope means the line will be steeper as it descends.

In our exercise, the equation \( y = 400 - 4x \) signifies a negative relationship. With a slope of -4, each increase of one unit in \( x \) results in a decrease of 4 units in \( y \). Thus, the graph shows a downward trajectory, confirming this negative correlation.