Problem 18 - Algebra 2

Question

Write an equation in slope-intercept form for the line that satisfies each set of conditions. passes through \((7,1)\) and \((7,8)\)

Step-by-Step Solution

Answer

The equation of the line is \( x = 7 \).

Step 1: Identify the Form of the Equation

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Calculate the Slope

The slope \( m \) is calculated by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).For the points \((7,1)\) and \((7,8)\), substitute:\[ m = \frac{8 - 1}{7 - 7} = \frac{7}{0} \]Since division by zero is undefined, this indicates a vertical line.

Step 3: Identify the Equation of the Vertical Line

For vertical lines, the equation is of the form \( x = c \), where \( c \) is the constant \( x \)-value for all points on the line. Since all points have an \( x \)-coordinate of 7, the equation of the line is \( x = 7 \).

Key Concepts

Slope Calculation

When finding the slope of a line, you're essentially measuring how steep the line is. The slope tells us how much the line rises vertically for each step it takes horizontally. This is calculated using the formula \[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here,

• \( y_2 \) and \( y_1 \) are the y-coordinates of two distinct points on the line.
• \( x_2 \) and \( x_1 \) are the x-coordinates of the same points.

To calculate the slope of a line, replace these variables with the coordinates from your points and solve the equation. In our example, the points each had the same x-coordinate, which made \( x_2 - x_1 = 0 \). This means that the line is vertical, rendering the slope undefined, as you cannot divide by zero.

Equation of a Line

The equation of a line gives us a complete description of a line on a graph. Typically, we use the slope-intercept form, \[ y = mx + b \] where

• \( m \) is the slope of the line
• \( b \) represents the y-intercept, or the value of y when x is zero.

If you know two points on a line, you can determine the equation of the line by calculating the slope and using one of the points to find the y-intercept.However, in the case of vertical lines, which have an undefined slope, the typical slope-intercept form does not work. Vertical lines are special: they have a constant x-value for every point.Thus, for vertical lines, we describe them with the equation \( x = c \), where \( c \) is the constant x-value of the line. This form makes it clear where the vertical line lies on the x-axis.

Vertical Lines

Vertical lines are unique in the world of linear equations. Unlike other lines, they don't tilt left or right. Instead, they go straight up and down. This makes their slope undefined because there's no horizontal change (i.e., the denominator in our slope formula \( x_2 - x_1 \) is zero).Vertical lines are best described by equations of the form:\[ x = c \]Here, \( c \) represents the fixed x-value for all points on the line, making it easy to remember that this line never crosses the x-axis.When dealing with vertical lines, always check your points to see if the x-coordinates are the same. If they are, it's a clear sign of a vertical line.These types of lines do not have a y-intercept because they run parallel to the y-axis and never cross it. Hence, the concept of slope-intercept form doesn't apply to them.