Problem 43 - College Algebra with Modeling and Visualization

Question

If a line passes through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right),\) then an equation of this line can be found by calculating the determinant. $$ \operatorname{det}\left[\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right]=0 $$ Find the standard form ax \(+b y=c\) of the line passing through the given points. $$ (6,-7) \text { and }(4,-3) $$

Step-by-Step Solution

Answer

The equation of the line is \(2x + y = 5\).

Step 1: Understand the Determinant Setup

The formula uses a determinant to calculate the line equation. The line passing through two points is set to a determinant equation:\[\operatorname{det}\left[\begin{array}{ccc}x & y & 1 \x_{1} & y_{1} & 1 \x_{2} & y_{2} & 1 \\end{array}\right]=0\] We need to replace \(x_1, y_1, x_2, y_2\) with the given coordinates of the points.

Step 2: Substitute Coordinates into the Determinant

Substitute the given points \((6, -7)\) and \((4, -3)\) into the determinant:\[\operatorname{det}\left[\begin{array}{ccc}x & y & 1 \6 & -7 & 1 \4 & -3 & 1 \\end{array}\right]=0\]

Step 3: Calculate the Determinant

Expand the determinant, which represents the area formed by these three points (should be zero for a line):\[x(-7 - (-3)) - y(6 - 4) + 1(6\cdot(-3) - (-7)\cdot 4) = 0\]Calculate each part: \(-7 - (-3) = -4\), \(6 - 4 = 2\), \(6\cdot(-3) = -18\), \((-7)\cdot 4 = -28\).

Step 4: Simplify the Determinant Expression

Simplify the equation:\[-4x - 2y + [-18 + 28] = 0\]Simplifying further:\[-4x - 2y + 10 = 0\]

Step 5: Rearrange to Standard Form ax+by=c

The goal is to arrange the equation into the form \(ax + by = c\). Start by simplifying:\[-4x - 2y = -10\]Divide through by \(-2\) to simplify:\[2x + y = 5\]

Step 6: Verification

Reassess to ensure the equation is correct and represents the line passing through both points. Substitute \((6, -7)\) and \((4, -3)\) to check. Both should satisfy \(2x + y = 5\), confirming the solution.

Key Concepts

Equation of a Line

An equation of a line is a mathematical statement that describes all the points that lie on a particular straight line in a coordinate plane. One of the most fundamental aspects of geometry, this equation helps in understanding how various points relate to one another along a path. When we talk about lines in two-dimensional space, an equation typically involves x and y coordinates, describing this relationship. In our given scenario, the equation is determined using two distinct points that lie on the line. These two points provide enough information to define the line completely.

A very flexible way to calculate the equation of a line is by employing the determinant method. By forming a matrix with the coordinates of the two points and a generic point \(x, y\), we can solve for when the determinant is zero. This calculation essentially finds the set of all \(x, y\) that satisfy the line through those two specific points.

Standard Form Equation

Standard form of a line, denoted as \(ax + by = c\), is a way to represent linear equations with integers and no fractions. This form has particular advantages, especially in coordinate geometry and algebraic operations.

One of the key benefits of using the standard form is its simplicity during some algebraic manipulations. It is also easy to transform into other useful forms like the slope-intercept form. In typical applications, the integer coefficients (\({a}\), \({b}\), and \({c}\)) make it more straightforward to handle equations when pairing with other algebraic systems. For example, in our exercise, the end goal was to derive this standard form from the determinant setup. After performing the determinant expansion and simplifying, we organized the expression into \(2x + y = 5\), showing how straightforward it can simplify into the standard form once solved and simplified correctly.

Coordinates

Coordinates are essential in geometry, providing a numerical way to represent points on a plane. The usage of coordinates allows for precise descriptions of location and movement within a given space. They are typically expressed as pairs \(x, y\), where \x\ controls the horizontal position and \y\ manages the vertical position.

In our example, two sets of coordinates, \(6, -7\) and \(4, -3\), were used to characterize points on the line. These coordinates are not just arbitrary numbers; they hold significant meaning and directly influence the line's expression we calculate. When using determinant methods or other means to find an equation of a line, coordinates play a critical role. They not only determine the plane's specific locations but also serve as crucial inputs for mathematical processes used in deriving equations that accurately reflect the geometric configuration of those points.