Problem 41 - Calculus Single Variable

Question

Sketch the line whose Cartesian equation is given. $$ x / 2=1 $$

Step-by-Step Solution

Answer

The line is vertical at \( x = 2 \).

Step 1: Rearrange the Equation

The given equation is \( \frac{x}{2} = 1 \). First, we multiply both sides of this equation by 2 to isolate \(x\). This gives us:\[ x = 2 \]

Step 2: Identify Characteristics of the Line

The equation \( x = 2 \) is a vertical line. It means that for any value of \( y \), \( x \) will always be 2. There is no slope since it's a vertical line.

Step 3: Plot Points and Draw the Line

To sketch the line, plot any points where \( x = 2 \). For example, draw a point at (2,1) and another at (2,-3) on a Cartesian coordinate plane. Draw a vertical line through these points.

Key Concepts

Vertical Line in Cartesian Plane

In the world of mathematics, particularly in geometry, a vertical line is incredibly distinctive. It is defined by the equation \( x = a \), where \( a \) is a constant. For any vertical line, all points share the same \( x \)-coordinate of \( a \). This means that the line stretches vertically, crossing the \( x \)-axis at this constant point.

Vertical lines have infinite length in the sense that they extend infinitely in both the upward and downward directions along the \( y \)-axis. They do not have a slope, as slope is defined by changes in \( y \) over changes in \( x \), which is impossible here since \( x \) does not change. Thus, the slope of a vertical line is considered undefined.

Understanding vertical lines is crucial when working with Cartesian equations. Be sure to recognize the direct and unwavering path a vertical line represents—it's the graphical embodiment of stability along the \( x \)-coordinate.

Understanding the Coordinate Plane

The coordinate plane is a fundamental concept in mathematics that provides a visual way to represent relationships between numbers through graphing. It is composed of two perpendicular lines or axes: the horizontal \( x \)-axis and the vertical \( y \)-axis. Together, they divide the plane into four quadrants.

Key points to understand about the coordinate plane:

• The point where the \( x \)-axis and \( y \)-axis intersect is the origin, denoted as (0,0).
• Any point on the coordinate plane is represented by an ordered pair \((x, y)\), where \( x \) is the horizontal position and \( y \) is the vertical position.
• Each axis can be positive or negative, reflecting the direction of the point from the origin.

The coordinate plane is essential for graphing lines, plotting points, and understanding spatial relationships. It provides context for the Cartesian equation we're discussing.

Equation Rearrangement Techniques

Rearranging an equation is a fundamental skill that allows you to alter the form of an equation while keeping equality intact. In the context of Cartesian equations, rearrangement often involves isolating a particular variable in order to identify the properties of the line it represents.

For example, consider the equation \(\frac{x}{2} = 1\). To understand the line it describes, we rearranged it to solve for \( x \):

• First, multiply both sides by 2 to eliminate the fraction.
• This simplifies to \( x = 2 \).

Through rearranging, we can clearly see that the equation describes a vertical line at \( x = 2 \). Correctly manipulating equations lets you draw meaningful and precise conclusions about the geometric or algebraic scenarios they depict.

Plotting Points and Sketching the Line

Plotting points is the act of marking specific locations on the coordinate plane, represented by ordered pairs \((x, y)\). It is fundamental to graphing the equation of a line. For a vertical line described by \( x = 2 \), any point can be plotted as long as its \( x \)-coordinate remains 2.

To sketch this line:

• Pick various \( y \)-coordinates, such as 1 or -3, and pair them with the consistent \( x \)-coordinate of 2. This gives points like (2,1) and (2,-3).
• Place these points on the coordinate plane.
• Draw a straight line passing through these plotted points, ensuring it extends vertically across the \( y \)-axis.

Plotting points offers clarity, allowing you to visualize the exact nature of the line in relation to the coordinate plane. Ensure to practice this method for accurate graphing.