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A horizontal line in the coordinate plane is a line that runs left to right and remains constant in the vertical direction. When you see a linear equation like \( y = -7 \), it means that the value of \( y \) is always \(-7\) regardless of the \( x \) value. This kind of equation is often called a constant function because it does not change as \( x \) changes.

To visualize this, imagine a straight line drawn parallel to the \( x \) axis. Every point on this line will share the same \( y \) coordinate, which, in our example, is \(-7\). The horizontal line will stretch across the graph, representing infinite solutions for various \( x \) values, but a single \( y \) value. This simplicity is why horizontal lines are easy to graph once you know the fixed \( y \)-value.

Solving linear equations involves finding the value of the variable that will make the equation true. These equations can take several forms, but they often appear as something that can be simplified or solved to isolate one variable.

Take for instance the equation given: \( y + 7 = 0 \). The goal is to solve for \( y \), which means isolating \( y \) by undoing any operations done to it. In this case:

• Subtract \( 7 \) from both sides of the equation: \( y + 7 - 7 = 0 - 7 \).
• This simplifies to \( y = -7 \).

Thus, the solution to the equation \( y + 7 = 0 \) is \( y = -7 \). Always remember, solving linear equations is about finding an equivalent equation where the variable is isolated and easy to interpret.

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which we can graph equations. It's defined by two perpendicular lines: the \( x \)-axis, which is horizontal, and the \( y \)-axis, which is vertical.

Each point on the coordinate plane is identified by an ordered pair \((x, y)\) that specifies its horizontal and vertical positions with respect to the axes. For example, the point \((3, -7)\) indicates a position 3 units to the right of the origin along the \( x \)-axis, and 7 units below the origin along the \( y \)-axis.

When graphing the equation \( y = -7 \), you create a horizontal line because the \( y \)-value remains constant at \(-7\) as the \( x \)-value changes. The line runs parallel to the \( x \)-axis and passes through all points where the \( y \) value is \(-7\), forming a clear, straight line that provides a visual representation of the solutions to the equation.