Problem 94 - Algebra for College Students

Question

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-1,6)\) and is parallel to the \(y\) -axis.

Step-by-Step Solution

Answer

x + 1 = 0

Step 1: Identify the Slope

Since the line is parallel to the y-axis, the slope is undefined. A line parallel to the y-axis is a vertical line.

Step 2: Determine the Equation Format for Vertical Lines

Vertical lines have the general equation in the format of \(x = k\), where \(k\) is the x-coordinate of any point on the line.

Step 3: Substitute the x-coordinate

The line passes through the point \((-1, 6)\). Since it is a vertical line, the equation is simply \(x = -1\).

Step 4: Write the Equation in Standard Form

To express the equation in standard form (Ax + By + C = 0), rewrite \(x = -1\) as \(x + 1 = 0\).

Key Concepts

standard form equation

The standard form of a linear equation is a way to express a line mathematically. It's written as \(Ax + By + C = 0\). Here, A, B, and C are integer coefficients, and x and y are variables representing any point on the line.
Standard form is especially useful because it provides a clear and concise way to represent a line.
It is also helpful for quickly identifying key features of the line, such as intercepts.
To convert any line equation to standard form, follow these steps:

• Move all terms to one side of the equation to set it to zero.
• Ensure A, B, and C are integers (if they’re not, you might need to multiply through by the least common multiple).
• A should be positive.

For example, an equation like \(y = 2x + 3\) can be rearranged to standard form as \(2x - y + 3 = 0\).

vertical line

Vertical lines have a unique property – their slope is undefined. This is because they run parallel to the y-axis and do not tilt.
A simple way to write the equation of a vertical line is using the format \(x = k\), where \(k\) is the x-coordinate of any point the line passes through.
In our exercise, the line passes through the point \((-1, 6)\). Since it is parallel to the y-axis, the equation is simply \(x = -1\).

This equation indicates that for any value of y, the x-coordinate remains -1.
Vertical lines are somewhat different from other lines because they don’t have the usual y-intercept form. Instead, their equation solely focuses on the x-coordinate.
Converting the vertical line equation to standard form is straightforward too. For \(x = -1\), you would rewrite it as \(x + 1 = 0\), ensuring it fits within \(Ax + By + C = 0\).

slope

The slope of a line measures its steepness and direction. Mathematically, slope (often denoted as \(m\)) is defined as the ratio of the change in y to the change in x between two points on the line:
\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\]
For non-vertical lines, the slope tells you how much y increases for a given increase in x. If \y\ increases as x increases, the slope is positive. If y decreases as x increases, the slope is negative.

However, when a line is vertical, its slope is undefined. This is because division by zero occurs (since \x\ does not change). Consider the line in our exercise, which is parallel to the y-axis. Any two points on this line will have the same x-coordinate, making \(x_2 - x_1=0\).
Thus, because vertical lines do not have a defined slope, we often use their specific equation form \x = k\ instead of dealing with slope calculations.