Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Use a determinant to find an equation of the line passing through the points. $$ (-4,3),(2,1) $$

Step-by-Step Solution

Answer

The equation of the line passing through points (-4,3) and (2, 1) is \(x + 3y = 5\)

Step 1: Calculate the Slope

The slope \(m\) of a line through points \((-4, 3)\) and \((2, 1)\) can be calculated using the determinant formula: \(m = (y_2 - y_1) / (x_2 - x_1)\). So, it would be \(m = (1 - 3) / (2 - (-4)) = -2 / 6 = -1/3\)

Step 2: Identify one point

Take one of the points as a reference. It can be either \((-4, 3)\) or \((2, 1)\). For this solution, let's use \((-4, 3)\). This means \(x_1 = -4\) and \(y_1 = 3\)

Step 3: Formulate the Equation of the Line

Now we can use the slope-intercept form of the equation of a line, which is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, \(x_1\) and \(y_1\) are the coordinates of one of the points. If we substitute the values we have \(y - 3 = -1/3 (x + 4)\). Multiply all terms by 3 to get rid of the fraction: \(3y - 9 = -1(x + 4)\). This simplifies to \(3y - 9 = -x - 4\). If we re-arrange to the conventional format we have \(x + 3y = 5\) as the equation of the line.

Key Concepts

Determinants

When dealing with linear equations and geometry, determinants simplify computations related to the coordinates of points. Essentially, a determinant is a scalar value derived from a square matrix. It provides useful properties for solving systems of linear equations. In the context of linear equations, specifically for finding the equation of a line, a determinant can help calculate the slope in certain instances, although in another context like three points forming a triangle, determinants are more crucial. Understanding that using matrix determinants is fundamental, you'll find them particularly useful when dealing with more complex geometric relationships.

Slope of a Line

The slope of a line is crucial as it indicates the line's steepness and direction. We calculate the slope using two points on the line. The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points.
• \(m\) represents the slope.

This calculation gives a ratio that indicates how much \(y\) increases or decreases as \(x\) increases. Positive slopes mean the line ascends from left to right, while negative slopes indicate it descends. In our exercise, the slope was found to be -1/3, meaning the line descends gently as \(x\) moves from left to right. The concept is essential in forming the line's equation and understanding its geometry.

Point-Slope Form

The point-slope form is a straightforward way to derive the equation of a line if you have one point on the line and the slope. The formula is: \[ y - y_1 = m(x - x_1) \] where:

• \((x_1, y_1)\) are the coordinates of the given point.
• \(m\) is the slope of the line.

This form is advantageous because it directly integrates the specific point, allowing you to easily form the line's equation without converting to the slope-intercept form initially, especially if a particular point on the line is known. In the exercise, using point \((-4, 3)\) and slope \(-1/3\), we applied this form to create the line equation before rearranging into the standard form, resulting in \(x + 3y = 5\). It makes understanding linear equations more accessible, especially in geometric contexts.