Problem 22 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((2,3)\) and \((0,-2)\)

Step-by-Step Solution

Answer

The equation of the line is \[ y = \frac{5}{2}x - 2 \].

Step 1 - Find the Slope

To determine the slope of the line that passes through points \(2, 3\) and \(0, -2\), use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Substitute the given points into the formula: \[ m = \frac{-2 - 3}{0 - 2} = \frac{-5}{-2} = \frac{5}{2} \].

Step 2 - Use the Point-Slope Form

Substitute the slope \( m = \frac{5}{2} \) and one of the points, such as \( (2, 3) \), into the point-slope form equation: \[ y - y_1 = m(x - x_1) \]. This gives: \[ y - 3 = \frac{5}{2}(x - 2) \].

Step 3 - Simplify the Equation

Distribute to simplify: \[ y - 3 = \frac{5}{2}x - 5 \]. Add 3 to both sides to get the equation in slope-intercept form: \[ y = \frac{5}{2}x - 2 \].

Key Concepts

Slope Formula

The slope of a line is crucial in understanding its direction and steepness. This concept is visually grasped by how 'tilted' the line appears on a graph. To find the slope between two points, we use the slope formula, written as \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. The variables \(x_1, y_1\) and \(x_2, y_2\) are coordinates of two different points the line passes through. Subtract the \(y\)-coordinates (vertical change), followed by the \(x\)-coordinates (horizontal change) and then divide the difference of \(y\) by the difference of \(x\). This gives us the 'rise over run' or the rate at which \(y\) changes with respect to \(x\). For example, given points \((2,3)\) and \((0,-2)\), applying the formula yields \[ m = \frac{-2 - 3}{0 - 2} = \frac{-5}{-2} = \frac{5}{2} \]. The positive slope indicates a line rising from left to right.

Point-Slope Form

Once we have the slope, we can write the equation of a line in the point-slope form. This form is especially useful when dealing with a known slope \(m\) and a point \(x_1, y_1\). The point-slope formula is expressed as, \[ y - y_1 = m(x - x_1) \]. It is derived from the definition of the slope and allows us to plug in one point directly.
Let’s use the previously found slope \( \frac{5}{2} \) and point \( (2, 3) \). Substituting these values, we get: \[ y - 3 = \frac{5}{2}(x - 2) \]. This linear equation directly relates a point to the general slope of the line.
In this form, transforming to the more standard slope-intercept form becomes simpler.

Slope-Intercept Form

The slope-intercept form of a line's equation is perhaps the most well-known representation. It is written as \[ y = mx + b \], where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis. This form is particularly favored for its clarity in immediately showing the slope and y-intercept, making graphing straightforward.
To convert our point-slope form equation \( y - 3 = \frac{5}{2}(x - 2) \) into slope-intercept form, first distribute the slope: \[ y - 3 = \frac{5}{2}x - 5 \].
Then, add 3 to both sides to isolate y: \[ y = \frac{5}{2}x - 2 \].
Here, the slope \(m = \frac{5}{2} \) and the y-intercept \( b = -2 \). This simplified format allows for an easier visualization and directly shows how the line behaves.