Problem 59 Describe what the graph of each ... [FREE SOLUTION] (2024)

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The coordinate plane is a two-dimensional surface formed by two number lines that intersect at a right angle. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis.
The point where these axes intersect is known as the origin, labeled as \((0,0 )\). The coordinate plane is used to graph equations and plot points using ordered pairs \( ( x, y ) \).

• The x-coordinate (first number in the pair) shows the position to the left or right of the origin.
• The y-coordinate (second number) shows the position above or below the origin.

Understanding how to navigate the coordinate plane is crucial for graphing any equation, including linear equations like the one in the exercise.

A horizontal line in the coordinate plane is a straight line that runs from left to right. This type of line is parallel to the x-axis and has a constant y-value.

• The equation of a horizontal line can be written in the form of \( y = c \), where \( c \) is a constant.
• This indicates that no matter what the value of \( x \) is, \( y \) remains the same.

For example, the equation \( y = -2 \) describes a horizontal line that crosses the y-axis at -2. To graph this line, simply draw a straight line through \( y = -2 \), extending it in both directions parallel to the x-axis.

Isolating variables is a crucial step in simplifying equations and solving for unknowns. In many cases, it means getting one variable by itself on one side of the equation.
For the exercise provided, the equation \( 3y = -6 \) needs to be simplified by isolating \( y \).

• First, divide every term in the equation by 3, yielding \( y = -2 \).
• This makes it clear that \( y \) is always equal to -2, which is a simpler and more recognizable form.

By isolating the variable first, it becomes easier to identify the form of the equation and understand how to graph it. In this case, isolating \( y \) revealed that the equation represents a horizontal line at \( y = -2 \).