Problem 7 - Algebra 1

Question

Write an equation of the line in point-slope form that passes through the given point and has the given slope. $$ (3,4), m=\frac{1}{2} $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the point (3, 4) and has the slope of 1/2 is \(y - 4 = 1/2 (x - 3)\).

Step 1: Identify the Given Point and Slope

The problem provides a point on the line, \((3,4)\), so \(x1 = 3\) and \(y1 = 4\). The slope is given as \(m = 1/2\).

Step 2: Insert Values into Point-Slope Form Equation

Substitute the known values into the point-slope form of the equation. Remember the form: \(y - y1 = m(x - x1)\). So in this case we have: \(y - 4 = 1/2 (x - 3)\).

Step 3: Simplify Equation

After substituting, the equation may initially appear complex, particularly because of the fractional slope. However, it does not need to be simplified further. The equation of the line is \(y - 4 = 1/2 (x - 3)\). This final equation was achieved by simply replacing the variables in the point-slope form equation with the given point and slope.

Key Concepts

Linear Equations

A linear equation is one of the most fundamental concepts in algebra. It represents a straight line when graphed on a coordinate plane. The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) represents the y-intercept, the point where the line crosses the y-axis.

In the context of the given exercise, a linear equation is required to express the relationship between \(x\) and \(y\) coordinates of all points on the line that passes through a specific point with a certain slope. Point-slope form is especially useful for this task because it provides a quick way to write down the equation by plugging in just one point \( (x_1, y_1) \) and the slope \(m\).

Slope of a Line

The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in the \(y\)-value (rise) to the change in the \(x\)-value (run) between two distinct points on the line. In mathematical terms, if you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \(m\) is defined as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

For the exercise provided, the slope \(m\) directly influences the angle at which the line tilts away from the horizontal axis. A slope of \( \frac{1}{2} \) means that for every unit increase in \(x\), \(y\) increases by half a unit. The sign of the slope also indicates the direction of the line: a positive slope (\(m > 0\)) suggests that the line rises as it moves from left to right, while a negative slope (\(m < 0\)) shows that it falls.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves the study of geometric figures using the coordinate plane—a grid defined by a horizontal axis (typically the \(x\)-axis) and a vertical axis (typically the \(y\)-axis). Points on this plane are described by ordered pairs of numbers, \( (x, y) \), which specify their positions relative to the two intersecting axes.

In the exercise, the point \( (3, 4) \) is an example of such an ordered pair, providing a specific location on the coordinate plane. This point, along with the slope, allows us to use the principles of coordinate geometry to determine the equation of a line. Furthermore, by employing the point-slope form, we not only define the line accurately but also utilize the visual and practical aspects of coordinate geometry to understand the spatial relationship between the line, the slope, and the points it passes through.