Problem 37 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (2,3),(4,3) $$

Step-by-Step Solution

Answer

The equation in slope-intercept form of the line passing through the points (2,3) and (4,3) is \(y = 3\).

Step 1: Compute the Slope

The formula to calculate the slope \(m\) between points \((x1,y1)\) and \((x2,y2)\) is \(m = (y2 - y1) / (x2 - x1)\). Apply the given points to the formula: \(m = (3 - 3) / (4 - 2) = 0 / 2 = 0\). So, the slope of the line is 0.

Step 2: Identify the y-intercept

The y-intercept is the point where the line crosses the y-axis. For a horizontal line, the y-intercept is the y-coordinate of any point on the line. Therefore, the y-intercept is 3.

Step 3: Write the Equation in Slope-Intercept Form

In the slope-intercept form equation \(y = mx + c\), replace \(m\) with 0 and \(c\) with 3, which results in the equation: \(y = 0x + 3\). Simplifying it, results in: \(y = 3\). This is the equation of the line in slope-intercept form.

Key Concepts

Equation of a Line

The equation of a line is a mathematical expression that represents a straight line. One of the most popular forms of an equation of a line is the slope-intercept form. This format is very straightforward for graphing and understanding linear relationships.

The slope-intercept form of a line is written as:

• \(y = mx + c\)

In this equation,:

• \(y\) is the dependent variable on the vertical axis, and \(x\) is the independent variable on the horizontal axis.
• \(m\) represents the slope of the line, which indicates how steep it is.
• \(c\) is the y-intercept, showing where the line crosses the y-axis.

Using this form makes it easy to visualize and plot lines, especially when comparing multiple lines. By knowing the slope and y-intercept, you can quickly derive the entire line on a graph.

Slope Calculation

Calculating the slope of a line is the first step when finding the slope-intercept form. The slope tells us how much the line inclines or declines. It's also an indication of how one variable changes with respect to another.

The formula to find the slope \(m\) between two points\((x_1, y_1)\) and \((x_2, y_2)\) is:

• \(m =\frac{y_2-y_1}{x_2-x_1}\)

This is a simple division: the change in \(y\) values (vertical change) over the change in \(x\) values (horizontal change).

This tells us how many units \(y\) changes for a one-unit change in \(x\). In the example, the slope was calculated between points \((2, 3)\) and \((4, 3)\). Since these points share the same \(y\)-value of 3, the slope \(m = 0\) indicating a horizontal line. A zero slope means there is no vertical change as \(x\) changes.

Y-intercept Determination

Finding the y-intercept is a key step in writing the equation of a line in slope-intercept form. The y-intercept, identified as \(c\) in the equation \(y = mx + c\), is the exact point where the line crosses the y-axis. This is when \(x\) equals zero.

For the equation of the line, this step is straightforward if you have any point on the line and the slope. For horizontal lines, like in our exercise, any point's \(y\)-coordinate can serve as the y-intercept. Thus in our example, the y-intercept is 3 because the line crosses the y-axis at \(y = 3\).

Once you have both the slope and y-intercept, you can combine them into the slope-intercept form, leading to a complete description of the line's behavior.