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The slope of a line measures how steep it is. It tells us how the line rises or falls as we move from left to right. In mathematical terms, the slope is represented by the letter \( m \) in the slope-intercept form equation \( y = mx + b \). It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

• If the slope is positive, the line rises as we move to the right.
• If the slope is negative, the line falls as we move to the right.
• If the slope is zero, the line is horizontal and has no vertical change.
• If the slope is undefined, the line is vertical.

The given exercise presents a line equation in the form \( 4x + 5y = 10 \). To determine the slope, we rearrange it to the slope-intercept form, \( y = -\frac{4}{5}x + 2 \). Here, \( m = -\frac{4}{5} \), meaning the line slopes downward. For every 5 steps you move horizontally to the right, you move 4 steps downwards.

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept equation form \( y = mx + b \), the y-intercept is represented by \( b \). It gives us a starting point on the graph, as it is the y-coordinate when the x-coordinate is zero.
To find the y-intercept from the equation \( 4x + 5y = 10 \), we first transform it into the slope-intercept form: \( y = -\frac{4}{5}x + 2 \). The y-coordinate where the line crosses the y-axis is 2. Hence, the y-intercept is the point (0, 2).
Understanding the y-intercept is crucial since it gives us a concrete point to start graphing. From this point, you can then use the slope to determine other points on the line. The y-intercept provides a visual anchor that makes graphing linear equations easier to manage.

Graphing a linear equation is essentially plotting the line on a coordinate plane. The objective is to understand visually how the line behaves across the plane. This can be done easily when you know both the slope and the y-intercept.
To start graphing, follow these steps:

• Identify the y-intercept from the equation. This is your initial point on the graph. For our example, the point is (0, 2).
• Determine the slope of the line. The slope \( -\frac{4}{5} \) tells you to move right by 5 units and then down by 4 units from the y-intercept to find the next point.
• Plot the next point on the graph, which would be (5, -2) given our slope and starting point.
• Draw a straight line through these two points. Extend it across the graph to illustrate the entire line.

Graphing these equations not only helps in visualizing where the line moves through the coordinate plane but also in understanding the equation's balance between the x and y variables. This capability is important when dealing with more complex systems of linear equations.