Problem 34 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ (2,3),(5,3) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = 3.

Step 1: Find the slope using the two points

To find the slope of the line between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Using the given points \((2,3)\) and \((5,3)\), we get: $$ m = \frac{3-3}{5-2} = \frac{0}{3} = 0 $$

Step 2: Find the y-intercept

Since we are given a point on the line, we can plug in its coordinates to find the y-intercept (b). Since the slope of the line is 0, the equation of the line takes the form: $$ y = 0 \cdot x + b $$ Substitute the coordinates of the given point \((2,3)\): $$ 3 = 0 \cdot 2 + b $$ Therefore, the y-intercept is \(b = 3\).

Step 3: Write the equation of the line in slope-intercept form

Now that we have the slope (m) and the y-intercept (b), we can write the equation of the line in slope-intercept form: $$ y = 0 \cdot x + 3 $$ Simplifying, we get the equation of the line: $$ y = 3 $$

Key Concepts

Slope Calculation

Slope calculation is the first step when writing the equation of a line in slope-intercept form. The slope (m) describes the steepness or incline of the line. It is a measure of how much the line rises or falls as you move from one point to another. To calculate the slope between two points

• Identify the coordinates of the points as \((x_1, y_1)\) and \((x_2, y_2)\).
• Apply the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For example, given the points \((2,3)\) and \((5,3)\), you subtract the y-coordinates and x-coordinates as shown: \( m = \frac{3-3}{5-2} \).
This leads to a slope of 0, which means the line is perfectly horizontal with no incline. Slope calculation is crucial because it dictates the line's direction and tilt in a graph.

Y-Intercept

The y-intercept (b) is another vital component in the slope-intercept form of a linear equation. The y-intercept is the location on the y-axis where the line crosses.For a line with the equation in slope-intercept form, this point is where

• the value of x = 0, leading to the equation \( y = mx + b \).

In our example problem, since the slope (m) is 0, the equation becomes \( y = b \) for all points on the line. By substituting one of the given points, such as \((2,3)\), into the equation: \( 3 = 0 \cdot 2 + b \), we solve to find that \( b = 3 \).
This calculation shows that regardless of the x-coordinate, the y-coordinate remains constant at 3 across the line, confirming the line is horizontal and intersects the y-axis at 3.

Linear Equations

A linear equation describes a straight line in two dimensions and is commonly expressed in the slope-intercept form, \( y = mx + b \), where:

• \( m \) represents the slope, indicating the line's tilt
• \( b \) represents the y-intercept, where the line crosses the y-axis.

In our task, we derived the linear equation from two points. When the slope \( m \) is 0, the equation simplifies to just \( y = b \), because no tilt exists, resulting in a horizontal line.
The exercise involved recognizing this special case of \( m = 0 \). This means all y-values are constant regardless of x, forming a horizontal line at \( y = 3 \).
Understanding linear equations not only allows us to graph lines but also aids in interpreting how changes in x affect y, recognizing different line types, such as horizontal, steep, or perfectly diagonal.