91Ó°ÊÓ

The concept of 'slope' is fundamental in understanding linear equations. The slope of a line measures its steepness and direction. It's represented by the letter \(m\) in the slope-intercept form \( y = mx + b \).
The slope is calculated as the ratio of the 'rise' (vertical change) to the 'run' (horizontal change) between two points on the line.
In simpler terms, slope tells us how much \(y\) changes for a given change in \(x\). Mathematically, it’s given by the formula: \ \ \frac{\text{Change in } y}{\text{Change in } x} or \frac{{y_2 - y_1}}{{x_2 - x_1}} \ In the given equation \( y = -x + 5 \), we compare it to the general form \( y = mx + b \). Here, \(m = -1\), which means:

• For every 1 unit increase in \(x\), \(y\) decreases by 1 unit.
• This negative value indicates a downward slope.

The y-intercept is where the line crosses the Y-axis. This tells us the value of \(y\) when \(x=0\). In the slope-intercept form \(y = mx + b\), \(b\) represents the y-intercept.
To find it, look at the constant term in the equation. For \(y = -x + 5\), the y-intercept \(b\) is \(5\).
This means the line crosses the Y-axis at the point \((0, 5)\).

• Always remember, \(b\) provides the starting point on the Y-axis for graphing the line.
• It's crucial for plotting the initial point when drawing the graph.

Graphing linear equations involves plotting points and drawing a line through them. Here's a step-by-step approach using the equation \(y = -x + 5\):

1. **Identify the y-intercept**: The y-intercept \(b = 5\) means the line crosses the Y-axis at \((0, 5)\). Plot this point.

2. **Use the slope to find another point**: The slope \(m = -1\) tells us the line goes down 1 unit for every 1 unit it goes right. Starting from \((0, 5)\), move down to \((1, 4)\).

3. **Draw the line**: Connect the plotted points with a straight line. This line extends infinitely in both directions.
Remember:

• The graph visually represents the equation.
• Always start with the y-intercept and use the slope for accuracy.
• Practice graphing to become comfortable with linear equations.