91Ó°ÊÓ

The slope of a line tells us how steep the line is. In simpler terms, it gives us the rate at which one variable changes in relation to another. For a linear function represented by the equation \( y = mx + b \), the slope is represented by 'm'.
Slope plays a critical role in determining the direction and angle of the linear graph.
A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
In our example, the slope is given as 3, indicating a rise of 3 units in the 'y' direction for every increase of 1 unit in the 'x' direction.
Always remember, the steeper the slope, the more vertical the line appears.

The y-intercept of a line is where the line crosses the y-axis. This occurs when the value of 'x' is 0.
The general form of a linear equation \( y = mx + b \) includes 'b' as the y-intercept.
In our exercise, to find the y-intercept, we substitute the coordinates of the given point into the equation. Given the line passes through \( (1, -6) \), substituting x = 1 and y = -6 into \( y = 3x + b \) helps find 'b'.
This process involves a few simple algebra steps: -6 = 3(1) + b.
Solving for b, we get \( -6 = 3 + b \) and thus \( b = -9 \).
Therefore, the y-intercept 'b' for our linear function is -9.
This y-intercept provides us an initial point from which we can draw the line on the graph.

Finding the y-intercept using coordinate substitution is a straightforward method.
It uses known points on a graph to determine the unknown value, typically 'b'.
In our exercise, we substitute the given point (1, -6) into the equation \( y = 3x + b \).
Plugging in x = 1 and y = -6, we get: \( -6 = 3(1) + b \).
This simplifies to -6 = 3 + b.
Now, solving for 'b', we subtract 3 from both sides to get \( b = -9 \).
Substitution makes it easy to find unknown values within the equation by using known coordinates, ensuring the final function accurately represents the given condition.
In this case, substituting the coordinates ensures our linear equation \( y = 3x - 9 \) is correct.