Problem 24 - Algebra 1

Question

Write an equation in point-slope form of the line that passes through the given points. $$ (4,-5),(-2,-7) $$

Step-by-Step Solution

Answer

The equation of the line in point-slope form that passes through the points (4,-5) and (-2,-7) is \(y = -\frac{1}{3}x - \frac{11}{3}\).

Step 1: Calculate the slope

Using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the provided points, calculate the slope of the line. Substituting \((x_1, y_1) = (4, -5)\) and \((x_2, y_2) = (-2, -7)\) into the formula, we will get \(m = \frac{-7 - (-5)}{-2 - 4} = \frac{2}{-6} = -\frac{1}{3}\).

Step 2: Using point-slope form

Substitute \(m\) and one set of coordinates \((x_1, y_1)\) into the formula for point-slope form. Here, \(m\) equals -1/3 from step 1, and let's use the point \((4, -5)\) as \((x_1, y_1)\). So the equation becomes \(y - (-5) = -\frac{1}{3}(x - 4)\) meaning \(y + 5 = -\frac{1}{3}x + \frac{4}{3}\).

Step 3: Simplify the equation

Convert the equation to simplified and more standard form. Solving the equation from step 2 to isolate y, we get \(y = -\frac{1}{3}x + \frac{4}{3} - 5 = -\frac{1}{3}x - \frac{11}{3}\).

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is key in coordinate geometry and forms an essential part of constructing linear equations. The slope represents how steep a line is and the direction it goes. It's calculated by the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Consider two points on a line, \((x_1, y_1)\) and \((x_2, y_2)\). The difference in the y-coordinates (rise) divided by the difference in the x-coordinates (run) gives us the slope, denoted by \(m\). Positive slope indicates an upward trend from left to right, whereas a negative slope indicates a downward trend. If the slope is zero, the line is horizontal, and if the slope is undefined (division by zero), the line is vertical.

For example, taking two points \((4, -5)\) and \((-2, -7)\), the slope is calculated as: \[ m = \frac{-7 - (-5)}{-2 - 4} = \frac{-7 + 5}{-2 - 4} = \frac{-2}{-6} = \frac{1}{3} \] which is a negative value showing a downward slope when moving from left to right.

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are used to represent straight lines and are commonly written in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, or the point where the line crosses the y-axis.

In the context of the exercise, after calculating the slope, the linear equation can be expressed in point-slope form, which relates the slope of a line and coordinates of a point on that line: \[ y - y_1 = m(x - x_1) \] This form is particularly useful for finding the equation of a line when we know a point on the line and the slope. Using the previously calculated slope \(-\frac{1}{3}\) and the point \((4, -5)\), the point-slope form equation is \[ y + 5 = -\frac{1}{3}(x - 4) \] which can be simplified to the slope-intercept form for easier interpretation and graphing.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system. This branch of geometry uses algebraic equations to represent geometric shapes and allows us to analyze the properties of shapes like lines, circles, and polygons with the help of coordinates.

In our solution, a pair of points have defined the position of a line on a coordinate plane. With these coordinates, we can start by calculating the slope and then proceed to form the equation of the line. This equation not only helps plot the line but also allows us to predict where the line will pass at a given x or y value.

Coordinate geometry is fundamental in the intersection of algebra and geometry, enabling us to solve real-world problems involving distances, midpoints, gradients, and areas, to name a few. Its applications are vast and integral to fields such as computer graphics, engineering, and navigation.