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Chapter 9: Problem 39

For Exercises \(37-40,\) graph. $$x=7$$

Short Answer

Expert verified
Based on the given step by step solution, the short answer is: To graph the equation \(x=7\), first recognize that it is a vertical line with a constant \(x\)-value. Plot points with an \(x\)-coordinate of 7, such as \((7, -2)\), \((7, 0)\), \((7, 2)\), and \((7, 4)\), on a coordinate plane. Connect these points with a straight edge to form the vertical line that passes through point \((7, 0)\) and is parallel to the \(y\)-axis. Label the line with the equation \(x=7\) to indicate that the graph represents this equation.

Step by step solution

01

Identify the type of equation

The given equation is \(x=7\). This is a vertical line equation, since the value of \(x\) is constant and the equation doesn't include a \(y\) variable.
02

Plot the line on a coordinate plane

Since the equation is \(x=7\), every point on this line will have an \(x\)-coordinate equal to \(7\). Thus, the line is a vertical line passing through the point \((7,0)\) and it has no slope, as it is parallel to the \(y\)-axis. Some points that lie on the line are: - \((7,-2)\) - \((7,0)\) - \((7,2)\) - \((7,4)\) Next, plot these points on a coordinate plane, making sure they line up vertically. Connect the points with a straight edge to form the vertical line.
03

Label the line

Finally, label the equation of the line, \(x=7\), near the line to indicate that the graph represents this equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Plane
The coordinate plane is a two-dimensional surface where we can map points, lines, and curves using a pair of numbers, also known as coordinates. It is composed of two number lines that intersect at a right angle. The horizontal line is called the 'x-axis', while the vertical line is the 'y-axis'. Their intersection point is known as the origin, which has coordinates (0,0). Each point on the plane is defined by an ordered pair (x, y), where 'x' represents the horizontal position, and 'y' represents the vertical position. When graphing, like the equation \(x=7\), we use this system to locate the exact position where our line will pass through on this grid.
Vertical Line Equation
A vertical line in the coordinate plane is one that goes straight up and down, parallel to the 'y-axis'. The equation of a vertical line is very simple as it takes the form \(x=a\), where 'a' represents a constant number. This means that no matter what the 'y' value (up or down the plane), 'x' will always be the same. For instance, the equation \(x=7\) indicates a vertical line that will cross the 'x-axis' at the point where \(x\) is 7, regardless of the 'y' value. This is the reason we won't find a 'y' variable in the equation of a vertical line - the 'x' value is fixed, and the line goes infinitely up and down.
Plotting Points
Plotting points is an essential skill when graphing equations on the coordinate plane. To plot a point, look at its coordinates, which are given in the form (x, y). Start at the origin (0, 0), move 'x' units along the 'x-axis', and then from there, move 'y' units along the 'y-axis'. If 'x' or 'y' is negative, you move in the opposite direction to the positive axis - left for negative 'x' values and down for negative 'y' values. When plotting the equation \(x=7\), it means every point you plot will have 'x' as 7 and 'y' as any value. This series of points, when connected, forms a vertical line.
X-Axis Intercept
An 'x-axis intercept' is a point where a graphed line crosses or 'intercepts' the 'x-axis'. In other words, it's where the value of 'y' is zero. For vertical lines, this concept is easy to grasp because they cross the 'x-axis' exactly once. Hence, the 'x-axis intercept' for the line \(x=7\) is the point (7, 0). It's the most direct evidence on a graph that a line represented by the equation \(x=7\) is indeed vertical - it doesn't vary horizontally and only comes into contact with the 'x-axis' at this single location.

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