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When we talk about **equations of lines**, particularly in the context of slope-intercept form, we're exploring a fundamental concept in algebra. The slope-intercept form is expressed as \( y = mx + b \). This equation provides a simple way to describe any straight line on a two-dimensional plane.

- **\( y \) represents the dependent variable**, determined by the independent variable \( x \), the slope \( m \), and the y-intercept \( b \).- **\( m \) denotes the slope of the line.** It tells us how steep or flat the line is. A positive slope means the line ascends from left to right, while a negative slope means it descends.- **\( b \) is the y-intercept.** This is where the line crosses the y-axis.
This form makes graphing lines easier and more intuitive, allowing us to quickly visualize the relationship between variables.

**Calculating slope** is essential to understanding how two points define a line. The slope \( m \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here is how it works:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.
• The numerator \( y_2 - y_1 \) represents the change in the y-value, also known as "rise."
• The denominator \( x_2 - x_1 \) indicates the change in the x-value, known as "run."

For example, between points \((8, 7)\) and \((-9, 7)\), substituting the values gives us a slope calculation of \( m = \frac{7 - 7}{-9 - 8} = \frac{0}{-17} = 0 \). This zero slope signifies a perfectly horizontal line, highlighting that there's no vertical change (rise) between the points.

The **y-intercept** is a key feature of a line described in slope-intercept form. Represented by \( b \) in the equation \( y = mx + b \), it indicates the point where the line crosses the y-axis.

- When a line intersects the y-axis, its \( x \) value is zero.- Therefore, the \( y \) value at this point is the y-intercept.

• A positive \( b \) positions the intersection above the origin.
• A negative \( b \) places it below the origin.
• If \( b = 0 \), the line passes through the origin itself.

For the horizontal line passing through \((8, 7)\) and \((-9, 7)\), we find that the y-intercept is 7. This is because both points share the same y-value, confirming that the line intersects the y-axis at \( y = 7 \). This intersection becomes the equation of the line: \( y = 7 \), illustrating that the y-value remains constant across all x-values.