91Ó°ÊÓ

Linear equations are fundamental in mathematics. It's an equation between two variables that gives a straight line when plotted on a graph. The general form of a linear equation is \(y = mx + b\), where:

• \(y\) is the dependent variable.
• \(x\) is the independent variable.
• m is the slope of the line.
• b is the y-intercept.

They are essential for understanding more complex concepts and appear in various real-world applications like predicting outcomes or modeling scenarios. Linear equations can be identified by the way they describe a relationship where a change in one variable is proportional to a change in the other variable. This relationship is graphically represented by a straight line in the coordinate plane, and different forms like \(y = mx + b\) and \(ax + by = c\) are often used in solving algebra problems.

Horizontal lines are a special kind of linear equation. They take the form \(y = c\), where \(c\) is a constant. These lines run straight across the graph from left to right. Here are some characteristics:

• They are parallel to the x-axis.
• They have a constant y-value.
• Their equation does not have an \(x\)-term.

In the problem, the equation \(y = -7\) is a horizontal line. This means all points on the line have the y-coordinate of -7, regardless of the value of \(x\). Therefore, no matter how far you move along the x-axis, the height (y-value) of the line remains the same.

The slope of a line measures its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, the slope \(m\) is given by:\[ m = \frac{\Delta y}{\Delta x} \]

• \(\Delta y\) represents the change in y-values between two points.
• \(\Delta x\) represents the change in x-values.

For horizontal lines, the slope is always 0 because the rise (\(\Delta y\)) is 0. The line does not move up or down as it progresses along the x-axis. Applying this to our problem, since the equation is \(y = -7\), the line is horizontal with a slope of 0. There's no change in y-values regardless of the change in x-values.