Problem 53 - Algebra and Trigonometry

Question

Determinants are used to write an equation of a line passing through two points. An equation of the line passing through the distinct points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by $$ \left|\begin{array}{lll} {x} & {y} & {1} \\ {x_{1}} & {y_{1}} & {1} \\ {x_{2}} & {y_{2}} & {1} \end{array}\right|=0 $$ Use this information to work Exercises \(53-54\) Use the determinant to write an equation of the line passing through \((3,-5)\) and \((-2,6) .\) Then expand the determinant, expressing the line's equation in slope-intercept form.

Step-by-Step Solution

Answer

The equation of the line passing through points \((3, -5)\) and \((-2, 6)\) is \(y = 11/2 * x + 7/2\).

Step 1: Substitute the given points into the determinant equation

Substitute points \((3, -5)\) and \((-2, 6)\) into the determinant equation, which results in: \[ \left|\begin{array}{ccc}x & y & 1 \3 & -5 & 1 \-2 & 6 & 1 \end{array}\right| = 0 \]

Step 2: Expand the determinant

Expanding the determinant into a sum of products, we get:'x*(-5)*1 + y*1*1 + 1*3*1 - 1*(-5)*(-2) - y*3*1 - x*1*6 = 0'. Simplify to get: '-5x+y+3-10-3y-6x = 0'

Step 3: Simplify and Rearrange

Simplify the equation by adding all terms related to x and y, and subtract 3 and 10 from both sides to isolate the other terms on the other side of the equation, which results in: '-11x - 2y -7 = 0'.

Step 4: Convert to slope-intercept form

Rearrange the equation into the slope-intercept form \(y = mx + b\). Do this by moving terms involving x to the other side, to give: \(y = 11/2 * x + 7/2\).

Key Concepts

Equation of a Line

The equation of a line is a fundamental concept in algebra and geometry. It provides a mathematical way to describe a straight line on a graph. To find the equation of a line, we often need two distinct points through which the line passes. These points can be represented as

• \((x_1, y_1)\)
• \((x_2, y_2)\)

Using the determinant method, these points can be placed into a 3x3 matrix. The determinant of this matrix equals zero, yielding the line's equation:The determinant is:\[\left|\begin{array}{ccc}x & y & 1 \ x_{1} & y_{1} & 1 \ x_{2} & y_{2} & 1 \end{array}\right|=0\]By solving this determinant, we derive the relationship between the variables x and y. This process is useful in coordinate geometry to establish the linear relationship given specific points.

Slope-Intercept Form

The slope-intercept form is a popular way to express the equation of a line. This form is represented as:\( y = mx + b \)where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept, where the line crosses the y-axis

The slope \( m \) tells us how steep the line is, and whether it is increasing or decreasing. In the equation \( y = \frac{11}{2}x + \frac{7}{2} \),\( \frac{11}{2} \) is the slope and \( \frac{7}{2} \) is the y-intercept. After expanding and simplifying a determinant, the equation may initially look different from the slope-intercept form. Converting it means rearranging the terms, making \( y \) the subject. This simplified form makes graphing much easier.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic formulas to represent geometric figures. It involves the use of the Cartesian coordinate system where every point is defined by an \(x\) and \(y\) coordinate. Understanding this system is crucial:

• The line equation is derived using these coordinates.
• Determinants are often employed to find relationships between points and lines.

When you work with the determinant of two points to find the line equation, you are applying coordinate geometry. It helps to visualize how points are positioned and related in a plane. This branch of mathematics is invaluable for solving problems involving lines, points, and shapes, and is a building block for more advanced topics like vector geometry.