Problem 31 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (5,-6),(5,-3) $$

Step-by-Step Solution

Answer

The equation of your line is \( x = 5 \).

Step 1: Visualize the Points

First imagine or draw on a piece of paper or on a graphing calculator the two points (5, -6) and (5, -3) which form a vertical line.

Step 2: Try to Determine the Slope

The slope between two points is normally determined by the formula \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \). For these points, the slope calculation would look like \( m = \frac{(-3 - (-6))}{(5 - 5)} \) = \( \frac{3}{0} \) this is undefined, because division by zero is undefined in mathematics.

Step 3: Find the Equation

When a line is vertical, the slope is undefined, and the equation can't be written in the traditional slope-intercept form \( y = mx + b \). Instead, the equation for vertical lines is simply \( x = c \) where \( c \) is the constant x-coordinate for all points on the line. For these points, that x-coordinate is 5, so the line's equation is \( x = 5 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to represent a straight line in algebra. It is written as:
\[ y = mx + b \]
where m represents the slope of the line and b is the y-intercept, which is the point where the line crosses the y-axis.

Understanding this form is essential because it allows us to easily identify the steepness and direction of a line, as well as to graph it quickly. The y represents any y-coordinate, m stands for the change in y (rise) over the change in x (run), and b provides the starting point of the line on the y-axis.

Graphing Points

Graphing points is foundational for understanding how to plot lines and curves on a coordinate plane. Each point is defined by an ordered pair, typically written as (x, y).
To graph a point, you locate the x-coordinate on the horizontal axis and the y-coordinate on the vertical axis. The spot where these two coordinates intersect is where the point is plotted.

Graphing the points before drawing the line through them helps to visualize the relationship between points and ensures accuracy when determining the line's slope and intercepts. It's the first critical step in connecting algebraic equations to geometric interpretations on a graph.

Undefined Slope

An undefined slope occurs when trying to calculate the slope of a vertical line. Remember, the slope m is a measure of how steep a line is, calculated as the change in y over the change in x.
However, for vertical lines, all points have the same x-coordinate, meaning the change in x is zero. Since division by zero is not possible, the slope is undefined.

Seeing an undefined slope is a signal that you're dealing with a vertical line, and therefore it cannot be represented by the slope-intercept form. Instead, you would use a different type of equation, the 'vertical lines equation'.

Vertical Lines Equation

A vertical line's equation is unique because it does not fit the slope-intercept form due to its undefined slope. The vertical line equation is simple:
\[ x = a \]
In this equation, a is the constant x-coordinate of all points on the line. The equation effectively means 'For any value of y, x will always be a.'

This unique representation is used for all vertical lines as they all have the same x-value, which makes them perpendicular to the x-axis and parallel to the y-axis. Whenever you encounter a pair of points with identical x-coordinates, as in our exercise example, you will employ this form to write their equation.